Research ArticleThe Cauchy Problem for a Weakly Dissipative 2-ComponentCamassa-Holm System

Sen Ming1 Han Yang1 and Yonghong Wu2

1 School of Mathematics Southwest Jiaotong University Chengdu 610031 China2Department of Mathematics and Statistics Curtin University Perth WA 6845 Australia

Correspondence should be addressed to Han Yang hanyang95263net

Received 18 November 2013 Accepted 14 December 2013 Published 3 February 2014

Academic Editor Shaoyong Lai

Copyright copy 2014 Sen Ming et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The weakly dissipative 2-component Camassa-Holm system is considered A local well-posedness for the system in Besov spaces isestablished by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation The wave-breakingmechanisms and the exact blow-up rate of strong solutions to the system are presented Moreover a global existence result forstrong solutions is derived

1 Introduction

We consider the following weakly dissipative 2-componentCamassa-Holm system

119906119905minus 119906

119909119909119905+ 2119896119906

119909+ 3119906119906

119909+ 120582119906

2119899+1

minus 1205731199062119898

119906119909119909

+ 120588120588119909= 2119906

119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120588119905+ (120588119906)

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120588 (119909 0) = 1205880(119909) 119909 isin R

(1)

where 119896 120582 120573 ge 0 are constants 119899 119898 are natural numbers(119909 119905) isin R times R+ and (119906

0 120588

0minus 1) isin 119861

119904

119901119903times 119861

119904minus1

119901119903with 119904 gt

max(32 1 + 1119901)In system (1) if 120582 = 120573 = 120588 = 0 we get the classical

Camassa-Holm equation [1]

119906119905minus 119906

119909119909119905+ 2119896119906

119909+ 3119906119906

119909= 2119906

119909119906119909119909

+ 119906119906119909119909119909

(2)

where 119906(119909 119905) is the fluid velocity in 119909 direction (or equiv-alently the height of the waterrsquos free surface above a flatbottom) and 119896 is a constant related to critical shallow waterwave speed The alternative derivation of (2) as a modelfor water waves can be found in Constantin and Lannes[2] Equation (2) models for the propagation of shallowwater waves have attracted attention of many researcherswith two remarkable features The first one is the presenceof solutions in the form of peaked solitary waves for 119896 = 0The peakon 119906(119909 119905) = 119888119890

minus|119909minus119888119905| with 119888 = 0 which is a featureobserved for the traveling waves of largest amplitude [3ndash6]The other feature is that the equation has breaking wavesIn other words the solutions remain bounded while theirslope becomes unbounded in finite time For 119896 gt 0 thesolitary waves are stable solitons [3ndash5 7] Li and Olver [8]not only obtained the local posedness but also gave theconditions which could lead to some solutions blowing upin finite time in Sobolev space 119867

119904 with 119904 gt 32 For othermethods to establish the local well-posedness and globalexistence of solutions to theCamassa-Holm equation or othershallow water models the reader is referred to [9ndash18] and thereferences therein

In general it is difficult to avoid the energy dissipationmechanisms in a real worldThus different types of solutions

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 801941 16 pageshttpdxdoiorg1011552014801941

2 Mathematical Problems in Engineering

for dissipative Camassa-Holm equation have been investi-gated For example Wu and Yin [19] studied the dissipativeCamassa-Holm equation

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120582 (119906 minus 119906

119909119909) = 2119906

119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

(3)

where 120582(119906 minus 119906119909119909

) (120582 gt 0) is the dissipative term Theyobtained the global existence result and blow-up results forstrong solutions in Sobolev space 119867

119904 with 119904 gt 32 byKatorsquos theory In [9] Lai andWu also investigated the weaklydissipative Camassa-Holm equation

119906119905minus 119906

119909119909119905+ 2119896119906

119909+ 3119906119906

119909+ 120582119906

2119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

(4)

where 119896 120582 120573 ge 0 are constants 119899119898 are natural numbersand 120582119906

2119899+1minus 120573119906

2119898119906119909119909

is the weakly dissipative term Theyobtained the local well-posedness in Sobolev space 119867

119904 with119904 gt 32 by using the pseudoparabolic regularization tech-nique and some estimates derived from the equation itselfand also developed a sufficient condition which guaranteedthe existence of weak solutions in Sobolev space 119867

119904 with1 lt 119904 le 32 We note that they only studied the equationin Sobolev space119867

119904On the other hand the Camassa-Holm equation also

admits many integrable multicomponent generalizations[20ndash32] For example

119898119905+ 2119896119906

119909+ 119906119898

119909+ 2119906

119909119898 + 120590120588120588

119909= 0

120588119905+ (120588119906)

119909= 0

(5)

where 119898 = 119906 minus 119906119909119909 The above system was derived in [33]

with 120590 = 1 In [20] Constantin and Ivanov gave a rigorousjustification of the system which is a valid approximation tothe governing equation for shallow water waves for 119896 = 0The 119906(119909 119905) represents the horizontal velocity of the fluid the120588(119909 119905) is related to the free surface elevation from equilibriumwith boundary assumptions and 119906 rarr 0 and 120588 rarr 1

as |119909| rarr infin They obtained the global well-posednesswith small initial data and the conditions for wave-breakingmechanism to the system For the case 120590 = minus1 it is notphysical as it corresponds to gravity pointing upwards and120588 rarr 0 as 119909 rarr infin The blow-up conditions are discussedin [25] For 119896 = 0 Gui and Liu [27] established the localwell-posedness for the system in a range of Besov spaces119861119904

119901119903times119861

119904minus1

119901119903with 119904 gt max(32 1+1119901)They also derivedwave-

breaking mechanisms and the exact blow-up rate for strongsolutions to the system in 119867

119904times 119867

119904minus1 with 119904 gt 32 Tian et al[34] obtained the local well-posedness for the system inBesov spaces 119861

119904

119901119903times 119861

119904

119901119903with 119904 gt max(32 1 + 1119901) They

also derived wave-breaking mechanisms for solutions anda result of blow-up solutions with certain profile in spaces119867119904times 119867

119904 with 119904 gt 32 Yan and Yin [21] investigated the 2-componentDegasperis-Procesi systemwhich is similar to the2-componentCamassa-Holm systemThey obtained the localwell-posedness in Besov spaces and derived a precise blow-up

scenario for strong solutions Guan and Yin [23] presented anew global existence result and several new blow-up resultsof strong solutions to an integrable 2-component Camassa-Holm shallow water system

In fact the 2-component Camassa-Holm system alsoadmits some generalizations due to the energy dissipationmechanisms in a real world In [35] Chen et al investigatedthe weakly dissipative 2-component Camassa-Holm system

119906119905minus 119906

119909119909119905+ 2119896119906

119909+ 3119906119906

119909minus 120588120588

119909+ 120582119906

2119899+1minus 120573119906

119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120588119905+ (120588119906)

119909= 0 119905 gt 0 119909 isin R

(6)

where 119896 120582 120573 ge 0 are constants and 119899 is a natural numberThey investigated the local well-posedness for the systemwithinitial data (119906

0 120588

0) isin 119867

119904times 119867

119904minus1 with 119904 ge 2 by Katorsquos theoryand derived a precise blow-up scenario for strong solutionsto the system

Motivated by the work in [9 21 23 27 33 35ndash38] westudy the weakly dissipative Camassa-Holm system (1) Wenote that the Cauchy problem of system (1) in Besov spaceshas not been discussed yet We state our main tasks withthree aspects Firstly we establish the local well-posednessof solutions to the system (1) Secondly we present theprecise blow-up criterions and exact blow-up rate for strongsolutions At last we derive a global existence result of strongsolutions Because of the presence of high order nonlinearterms 1199062119899+1 and 119906

2119898119906119909119909 the system (1) loses the conservation

law 119864 = intR(119906

2+ 119906

2

119909)d119909 which plays an important role in

studying system (1)Now we rewrite system (1) as

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1

2

1205882]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

2119898119906119909119909

]

119905 gt 0 119909 isin R

120588119905+ (120588119906)

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120588 (119909 0) = 1205880(119909) 119909 isin R

(7)

where the operators 119875(119863) = minus120597119909(1 minus 120597

119909119909)minus1 and 119875

1(119863) = (1 minus

120597119909119909

)minus1 We define the space

119864119904

119901119903(119879)

=

119862([0 119879] 119861119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) 1 le 119903 lt infin

119871infin

([0 119879] 119861119904

119901infin) cap lip ([0 119879] 119861

119904minus1

119901infin) 119903 = infin

(8)

with 119879 gt 0 119904 isin R 119901 isin [1infin] and 119903 isin [1infin]Themain results of this paper are stated as follows Firstly

we present the local well-posedness theorem

Mathematical Problems in Engineering 3

Theorem 1 For 1 le 119901 119903 le infin and 119904 gt max(32 1 + 1119901)Let (119906

0 120588

0minus 1) isin 119861

119904

119901119903times 119861

119904minus1

119901119903 There exists a time 119879 gt 0 such

that the initial value problem (1) has a unique solution (119906 120588 minus

1) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) and the map (119906

0 120588

0minus 1) rarr (119906 120588 minus

1) is continuous from a neighborhood of (1199060 120588

0minus 1) in 119861

119904

119901119903times

119861119904minus1

119901119903into 119862([0 119879] 119861

1199041015840

119901119903) cap 119862

1([0 119879] 119861

1199041015840minus1

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) cap

1198621([0 119879] 119861

1199041015840minus2

119901119903) for every 119904

1015840lt 119904 when 119903 = +infin and 119904

1015840= 119904

whereas 119903 lt +infin

We obtain the following blow-up results

Theorem 2 Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with 119904 gt 52 and119879 gt 0 be the maximal existence time of the solution (119906 120588) tosystem (1) with initial data (119906

0 120588

0) Then the corresponding

solution blows up in finite time if and only if

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909) = minusinfin

119900119903

lim119905rarr119879

minus

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin

(9)

Theorem 3 Let 119898 = 0 in system (1) Assume (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the initial value (1199060 120588

0) satisfies

1199060119909(119909

0) lt minus120573 minus radic2119870 where 119870 is a fixed constant defined in

(95) and 1205880(119909

0) = 0 with the point 119909

0defined by 119906

0119909(119909

0) =

inf119909isinR1199060119909(119909) Then the corresponding solution (119906 120588 minus 1) to

system (1) with 119898 = 0 blows up in finite time Namely thereexists a 119879

0with 0 lt 119879

0le minus2(1 minus 120575)(119906

0119909(119909

0) + 120573) such that

lim inf119905rarr119879

minus

0

inf119909isinR

119906119909(119905 119909) = minusinfin (10)

where 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870

Theorem 4 Let 119896 = 0 in system (1) Assume (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and the initial value (1199060 120588

0) satisfies

that 1199060is odd 120588

0is even and 119906

0119909(0) lt minusradic2119870

1 where 119870

1

is a fixed constant defined in (109) and 1205880(0) = 0 Then the

corresponding solution (119906 120588 minus 1) to system (1) with 119896 = 0

blows up in finite time More precisely there exists a 1198791with

0 lt 1198791le minus2(1 minus 120575

1)119906

0119909(0) such that

lim inf119905rarr119879

minus

1

119906119909(119905 0) = minusinfin (11)

In addition if 1205880119909(119909

0) = 0 with some 119909

0isin R satisfying

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) then there exists a 1198792with 0 lt 119879

2le

minus(2((1 minus 1205751)119906

0119909(0))) such that

(i) lim suptrarrTminus2

supxisinR

120588x (t x) = +infin if 1205880x (x0) gt 0

(ii) lim inf119905rarr119879

minus

2

inf119909isinR

120588119909(119905 119909) = minusinfin 119894119891 120588

0119909(119909

0) lt 0

(12)

where 1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1

We also obtain the exact blow-up rate of strong solutionsto system (1)

Theorem 5 Let 119879 lt infin be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) with119898 = 0 Theinitial data (119906

0 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with 119904 gt 32 satisfies1199060119909(119909

0) lt minus120573 minus radic2119870

2 where 119870

2is a fixed constant defined

in (120) and 1205880(119909

0) = 0 with the point 119909

0defined by 119906

0(119909

0) =

inf119909isinR1199060119909(119909) then

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (13)

Now we present a global existence result of strongsolutions to system (1)

Theorem6 Let119898 = 0 in system (1) and (1199060 120588

0minus1) isin 119867

2times119867

1If120588

0(119909) = 0 for all119909 isin R then the corresponding strong solution

(119906 120588 minus 1) to system (1) with 119898 = 0 exists globally in time

The remainder of this paper is organized as followsIn Section 2 some properties of Besov space and a prioriestimates for solutions of transport equation are reviewedSection 3 is devoted to the proof of Theorem 1 The proofs ofwave-breaking results and the precise blow-up rate of strongsolutions to system (1) are given in Section 4 respectivelyTheproof of Theorem 6 is presented in Section 5

Notation In this paper we denote the convolution onR by lowast

the norm of Lebesgue space 119871119901 by sdot

119871119901 1 le 119901 le infin and the

norm in Sobolev space 119867119904 119904 isin R by sdot

119867119904 and the norm in

Besov space119861119904119901119903

119904 isin R by sdot119861119904

119901119903

Here we denote 119886+ = 119886+120576where 120576 gt 0 is a sufficiently small number

2 Preliminary

In this section we recall some basic facts in Besov space Onemay check [21 39ndash42] for more details

Proposition 7 (see [40 42]) Let 119904 isin R 1 le 119901 and 119903 le +infinThe nonhomogeneous Besov space is defined by 119861

119904

119901119903= 119891 isin

1198781015840(R119899

) | 119891119861119904

119901119903

lt +infin where

10038171003817100381710038171198911003817100381710038171003817119861119904

119901119903

=

(

infin

sum

119895=minus1

211989511990311990410038171003817

100381710038171003817Δ119895119891

10038171003817100381710038171003817

119903

119871p)

1119903

119903 lt infin

sup119895geminus1

211989511990410038171003817100381710038171003817Δ119895119891

10038171003817100381710038171003817119871119901 119903 = infin

(14)

Moreover 119878119895119891 = sum

119895minus1

119902=minus1Δ119902119891

Proposition 8 (see [40 42]) Let 119904 isin R 1 le 119901 119903 119901119895 119903119895le infin

and 119895 = 1 2 then consider the following

(1) Embedding 11986111990411990111199031

997893rarr 119861119904minus119899(1119901

1minus11199012)

11990121199032

if 1199011

le 1199012and

1199031le 119903

2 And 119861

1199042

1199011199032

997893rarr 1198611199041

1199011199031

locally compact if 1199041le 119904

2

(2) Algebraic properties for any 119904 gt 0 119861119904

119901119903cap 119871

infin is analgebra 119861119904

119901119903is an algebra hArr 119861

119904

119901119903997893rarr 119871

infinhArr 119904 gt

119899119901 119900119903 119904 ge 119899119901 119886119899119889 119903 = 1

4 Mathematical Problems in Engineering

(3) Complex interpolation

100381710038171003817100381711989110038171003817100381710038171198611205791199041+(1minus120579)1199042

119901119903

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

1198611199041

119901119903

10038171003817100381710038171198911003817100381710038171003817

1minus120579

1198611199042

119901119903

119891119900119903 119890V119890119903119910 119891 isin 1198611199041

119901119903cap 119861

1199042

119901119903 120579 isin [0 1]

(15)

(4) Fatoursquos Lemma if (119906119899)119899isinN is bounded in119861

119904

119901119903and 119906

119899rarr

119906 in 1198781015840(R) then 119906 isin 119861

119904

119901119903and

119906119861119904

119901119903

le lim119899rarrinfin

inf 1003817100381710038171003817119906119899

1003817100381710038171003817119861119904

119901119903

(16)

(5) Let 119898 isin R and 119891 be an 119878119898-multiplier (ie 119891 R119899

rarr

R is smooth and satisfies that for every 120572 isin N119899 thereexists a constant 119862

120572such that |120597

120572119891(120585)| le 119862

120572(1 +

|120585|)119898minus|120572| for all 120585 isin R119899) Then the operator 119891(119863) is

continuous from 119861119904

119901119903to 119861

119904minus119898

119901119903

(6) Density 119862infin

119888is dense in 119861

119904

119901119903hArr 1 le 119901 119903 lt infin

Lemma 9 (see [40]) Let 119868 be an open interval of R Let 119904 gt 0

and 1199041be the smallest integer such that 119904

1ge 119904 Let 119865 119868 rarr R

satisfy 119865(0) = 0 and 1198651015840isin 119882

1199041infin

(119868R) Assume that V isin 119861119904

119901119903

has values in 119869 subsub 119868 Then 119865(V) isin 119861119904

119901119903and there exists a

constant 119862 depending only on 119904 119868 119869119873 such that

119865 (V)119861119904

119901119903

le 119862(1 + V119871infin)

1199041100381710038171003817100381710038171198651015840100381710038171003817100381710038171198821199041infin(119868)

V119861119904

119901119903

(17)

Lemma 10 (see [40]) Let 119868 be an open interval of R Let119904 gt 0 and 119904

1be the smallest integer such that 119904

1ge 119904 Let

119865 119868 rarr R satisfy 1198651015840(0) = 0 and 119865

10158401015840isin 119882

1199041infin

(119868R) Assumethat 119906 V isin 119861

119904

119901119903has values in 119869 subsub 119868 Then there exists a

constant 119862 depending only on 119904 119868 119869119873 such that

119865 (119906) minus 119865 (V)119861119904

119901119903

le 119862(1 + 119906119871infin)

11990411003817100381710038171003817100381711986510158401015840100381710038171003817100381710038171198821199041infin(119868)

times (119906 minus V119861119904

119901119903

sup120579isin[01]

120579119906 + (1 minus 120579) V119871infin

+ 119906 minus V119871infin sup120579isin[01]

120579119906 + (1 minus 120579) V119861119904

119901119903

)

(18)

Lemma 11 (see [40]) Assume that 1 le 119901 and 119903 le infin thefollowing estimates hold

(i) for 119904 gt 0 then

1003817100381710038171003817119891119892

1003817100381710038171003817119861119904

119901119903

le 119862(10038171003817100381710038171198911003817100381710038171003817119861119904

119901119903

10038171003817100381710038171198921003817100381710038171003817119871infin +

10038171003817100381710038171198911003817100381710038171003817119871infin

10038171003817100381710038171198921003817100381710038171003817119861119904

119901119903

)

(19)

(ii) for 1199041

le 1119901 1199042

gt 1119901 (1199042

ge (1119901) if 119903 = 1) and1199041+ 119904

2gt 0 then

1003817100381710038171003817119891119892

10038171003817100381710038171198611199041

119901119903

le 119862100381710038171003817100381711989110038171003817100381710038171198611199041

119901119903

100381710038171003817100381711989210038171003817100381710038171198611199042

119901119903

(20)

where 119862 are constants independent of 119891 119892

Then we present two related lemmas for the followingtransport equations

119891119905+ 119889 sdot nabla119891 = 119865

1198911003816100381610038161003816119905=0

= 1198910

(21)

where 119889 R times R119899rarr R119899 stands for a given time dependent

vector field and 1198910 R119899

rarr R119898 and 119865 R times R119899rarr R119898 are

known data

Lemma 12 (see [39 40]) Let 1 le 119901 le 1199011le infin 1 le 119903 le infin

and 1199011015840= 119901(119901 minus 1) Assume that 119904 gt minus119899 sdot min(1119901

1 1119901

1015840)

or 119904 gt minus1 minus 119899 sdot min(11199011 1119901

1015840) if nabla sdot 119889 = 0 Then there

exists a constant 119862 depending only on 119899 119901 1199011 119903 119904 such that

the following estimate holds true

10038171003817100381710038171198911003817100381710038171003817infin

119905([0119905]119861

119904

119901119903)

le 1198901198621int119905

0119885(120591)d120591

times [10038171003817100381710038171198910

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890minus1198621int120591

0119885(120585)119889120585

119865 (120591)119861119904

119901119903

d120591]

(22)

with

119885 (119905)

=

nabla119889 (119905)1198611198991199011

1199011infincap119871infin 119904 lt 1 +

119899

1199011

nabla119889 (119905)119861119904minus1

1199011119903

119904 gt 1+

119899

1199011

119900119903 119904=1+

119899

1199011

119903=1

(23)

If 119891 = 119889 then for all 119904 gt 0 (nabla sdot 119889 = 0 119904 gt minus1) (22) holds with119885(119905) = nabla119889(119905)

119871infin

Let us state the existence result for transport equationwith data in Besov space

Lemma 13 (see [40]) Let 119901 1199011 119903 119904 be as in the statement

of Lemma 12 and 1198910

isin 119861119904

119901119903and 119865 isin 119871

1([0 119879] 119861

119904

119901119903) 119889 isin

119871120588([0 119879] 119861

minus119872

infininfin) is a time dependent vector field for some

120588 gt 1119872 gt 0 such that if 119904 lt 1 + 1198991199011then nabla119889 isin

1198711([0 119879] 119861

1198991199011

1199011infin

cap 119871infin) if 119904 gt 1 + 119899119901

1or 119904 = 1 + 119899119901

1 119903 = 1

then nabla119889 isin 1198711([0 119879] 119861

119904minus1

1199011119903) Thus transport equation (21) has a

unique solution 119891 isin 119871infin([0 119879] 119861

119904

119901119903)⋂(⋂

1199041015840lt119904

119862([0 119879] 1198611199041015840

1199011))

and (22) holds true If 119903 lt infin then one has 119891 isin 119862([0 119879] 119861119904

119901119903)

3 The Proof of Theorem 1

We finish the proof of Theorem 1 by the following steps

Mathematical Problems in Engineering 5

31 Existence of Solutions For convenience we denote 120578 =

120588 minus 1 and rewrite (7) as

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205782

2

+ 120578]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

2119898119906119909119909

]

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909= minus119906

119909120578 minus 119906

119909 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(24)

We use a standard iterative process to construct the approxi-mate solutions to (24)

Step 1 Starting from 1199060

= 1205780

= 0 we define by inductiona sequence of smooth functions (119906

119894 120578

119894)119894isinN isin 119862(R+

119861infin

119901119903)2

solving the following transport equation

(120597119905+ 119906

119894120597119909) 119906

119894+1= 119865 (119905 119909) 119905 gt 0 119909 isin R

(120597119905+ 119906

119894120597119909) 120578

119894+1= minus120578

119894120597119909119906119894minus 120597

119909119906119894 119905 gt 0 119909 isin R

119906119894+1

(119909 0) = 119906119894+1

0(119909) = 119878

119894+11199060 119909 isin R

120578119894+1

(119909 0) = 120578119894+1

0(119909) = 119878

119894+11205780 119909 isin R

(25)

where

119865 (119905 119909) = 119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+ 120578119894]

+ 1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

(26)

Since all the data 119878119894+1

1199060 119878

119894+11205780isin 119861

infin

119901119903 Lemma 13 enables us

to show that for all 119894 isin N the system (25) has a global solutionwhich belongs to 119862(R+

119861infin

119901119903)2

Step 2 Next we prove that (119906119894 120578119894)119894isinN is uniformly bounded

in 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

According to Lemma 12 we have the following inequali-ties for all 119894 isin N10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

le 119890

1198621int119905

0119906119894(120591)119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890

minus1198621int120591

0119906119894(120585)119861119904

119901119903

d120585119865 (120591 sdot)

119861119904

119901119903

d120591]

(27)

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

1198622int119905

0119906119894119861119904

119901119903

d120591[10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198622int120591

0119906119894119861119904

119901119903

d120585

times

10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

d120591]

(28)

From Proposition 8 Lemma 11 and 119878minus1-multiplier property

of 119875(119863) it yields that1003817100381710038171003817100381710038171003817

119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+120578119894]

1003817100381710038171003817100381710038171003817119861119904

119901119903

le 119862(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

119861119904

119901119903

+2119896

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817

2

119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

(29)

Using the 119878minus2-multiplier property of 119875

1(119863) we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862

100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

(30)

(i) By the assumption of 119891(119906119894) = (119906

119894)2119899+1 and Lemma 9

we have100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119871infin)

(119904minus1)110038171003817100381710038171003817119891101584010038171003817100381710038171003817119882(119904minus1)1infin

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

(119904minus1)11003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(31)

(ii) As we know if max(32 1 + 1119901) lt 119904 le 2 + 1119901 then119861119904minus1

119901119903is an algebra if 119904 gt 2+1119901 then119861

119904minus2

119901119903is an algebra

By the assumption of ℎ(119906119894) = (119906119894)2119898 and Lemmas 10

and 11 we deduce that100381710038171003817100381710038171003817

120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862120573

100381710038171003817100381710038171003817

(119906119894)

2119898100381710038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

(119904minus1)1

times

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(32)

6 Mathematical Problems in Engineering

Combining (i) and (ii) yields100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(33)

Thanks to Lemma 11 we get10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(34)

Therefore from (27) to (34) we obtain10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 1198623sdot 119890

1198623int119905

0119906119894119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198623int120591

0119906119894119861119904

119901119903

d120585(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 119896 + 1)

1199041

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)] d120591

(35)

Let us choose a 119879 gt 0 such that 211990411198621199041+1

3( 119906

119894

0119861119904

119901119903

+ 120578119894

0119861119904minus1

119901119903

+

119896 + 1)1199041119879 lt 1 and

(119896 + 1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(36)

Inserting (36) into (35) yields

(119896 + 1 +

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(37)

Therefore (119906119894 120578119894)119894isinN is uniformly bounded in119862([0 119879] 119861

119904

119901119903)times

119862([0 119879] 119861119904minus1

119901119903) From Proposition 8 Lemma 11 and the fol-

lowing embedding properties

119861119904

119901119903997893rarr 119861

119904minus1

119901119903 119861

119904minus1

119901119903997893rarr 119861

119904minus2

119901119903 (38)

we have10038171003817100381710038171003817119906119894120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

(39)

We deduce that 119906119894120597119909119906119894+1 and 119865(119905 119909) are uniformly bounded

in 119862([0 119879] 119861119904minus1

119901119903) in the same way that 119906

119894120597119909120578119894+1

minus120578119894120597119909119906119894minus

120597119909119906119894 are uniformly bounded in 119862([0 119879] 119861

119904minus2

119901119903) Using the

system (25) we have (120597119905119906119894+1

120597119905120578119894+1

) isin 119862([0 119879] 119861119904minus1

119901119903) times

119862([0 119879] 119861119904minus2

119901119903) is uniformly bounded which derives that

(119906119894 120578

119894)119894isinN is uniformly bounded in 119864

119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

Step 3 Now we demonstrate that (119906119894 120578

119894)119894isinN is a Cauchy

sequence in 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

In fact according to (25) we note that for all 119894 119895 isin N wehave

(120597119905+ 119906

119894+119895120597119909) (119906

119894+119895+1minus 119906

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909119906119894+1

+ 119875 (119863)

times [(119906119894+119895

minus 119906119894) (119906

119894+119895+ 119906

119894) +

1

2

120597119909(119906

119894+119895minus 119906

119894)

sdot 120597119909(119906

119894+119895+ 119906

119894) + 2119896 (119906

119894+119895minus 119906

119894) +

1

2

(120578119894+119895

minus 120578119894)

times (120578119894+119895

+ 120578119894) + (120578

119894+119895minus 120578

119894) ]

+ 1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)

+ 120573 ((119906119894+119895

)

2119898

(119906119894+119895

)119909119909

minus (119906119894)

2119898

(119906119894)119909119909

)]

(40)

(120597119905+ 119906

119894+119895120597119909) (120578

119894+119895+1minus 120578

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909120578119894+1

minus (120578119894+119895

minus 120578119894) 120597

119909119906119894

minus 120578119894+119895

120597119909(119906

119894+119895minus 119906

119894) minus 120597

119909(119906

119894+119895minus 119906

119894)

(41)

(1) We estimate the terms in the right side of (40) ByLemma 11 we have10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (119906

119894+119895minus 119906

119894) (119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894+119895

+ 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [120597

119909(119906

119894+119895minus 119906

119894) 120597

119909(119906

119894+119895+ 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [2119896 (119906

119894+119895minus 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862119896

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817119875 (119863) [(120578

119894+119895minus 120578

119894) (120578

119894+119895+ 120578

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+119895

+ 12057811989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (120578

119894+119895minus 120578

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

(42)

Mathematical Problems in Engineering 7

We use Lemma 10 to estimate the high order terms in (40)Firstly we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

119904110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(43)

Secondly by 1199062119898

119906119909119909

= 120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909and

Lemma 11 we deduce that100381710038171003817100381710038171003817

120573 ((119906119894+119895

)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

le 119862(

100381710038171003817100381710038171003817

120597119909((119906

119894+119895)

2119898

119906119894+119895

119909minus (119906

119894)

2119898

119906119894

119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

+

100381710038171003817100381710038171003817

(119906119894+119895

)

2119898minus1

(119906119894+119895

119909)

2

minus (119906119894)

2119898minus1

(119906119894

119909)

2100381710038171003817100381710038171003817119861119904minus3

119901119903

)

(44)

Hence100381710038171003817100381710038171003817

1198751(119863) 120573 [(119906

119894+119895)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

1199041+110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(45)

(2) Now we estimate the right side of (41)10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817(120578

119894+119895minus 120578

119894) 120597

11990911990611989410038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

1003817100381710038171003817100381712059711990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120578119894+119895

120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(46)

Combining (1) (2) and Lemma 12 yields for all 119905 isin [0 119879]

10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817119906119894+119895+1

0minus 119906

119894+1

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862

times int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)(

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 1)

+

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+ 119896 + 1)

1199041+1

] d120591

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817120578119894+119895+1

0minus 120578

119894+1

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+ 1)

+(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

] d120591

(47)

Since (119906119894 120578119894)119894isinN is uniformly bounded in119864

119904

119901119903(119879)times119864

119904minus1

119901119903(119879) and

119906119894+119895+1

0minus 119906

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021199060

120578119894+119895+1

0minus 120578

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021205780

(48)

there exists a constant 119862119879independent of 119894 119895 such that for all

119905 isin [0 119879] we get10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862119879[2

minus119894+ int

119905

0

(

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

) d120591]

(49)

By induction we obtain that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le

(119862119879119879)

119894+1

(119894 + 1)

[

1003817100381710038171003817100381711990611989510038171003817100381710038171003817119871infin([0119879]119861

119904minus1

119901119903)+

1003817100381710038171003817100381712057811989510038171003817100381710038171003817119871infin([0119879]119861

119904minus2

119901119903)]

+ 119862119879

119894

sum

119897=0

2minus(119894minus119897)

(119862119879119879)

119897

119897

(50)

Since 119906119895119871infin

119879(119861119904

119901119903) 120578

119895119871infin

119879(119861119904minus1

119901119903)are bounded independently of

119895 we conclude that there exists a new constant 1198621198791

such that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le 1198621198791

2minus119894

(51)

Consequently (119906119894 120578119894)119894isinN is a Cauchy sequence in 119862([0 119879]

119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

8 Mathematical Problems in Engineering

Step 4 We end the proof of existence of solutions

Firstly since (119906119894 120578

119894)119894isinN is uniformly bounded in

119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) according to the Fatou

property for Besov space it guarantees that (119906 120578) alsobelongs to 119871

infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903)

Secondly for (119906119894 120578

119894)119894isinN is a Cauchy sequence in

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) so it converges to some

limit function (119906 120578) isin 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) An

interpolation argument insures that the convergence holdsin 119862([0 119879] 119861

1199041015840

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904 It is easy to

pass to the limit in (25) and to conclude that (119906 120578) is indeeda solution to (24) Thanks to the fact that (119906 120578) belongs to119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) we know that the right

side of the first equation in (24) belongs to 119871infin([0 119879] 119861

119904

119901119903)

and the right side of the second equation in (24) belongs to119871infin([0 119879] 119861

119904minus1

119901119903) In the case of 119903 lt infin applying Lemma 13

derives (119906 120578) isin 119862([0 119879] 1198611199041015840

119901119903) times119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904

Finally with (24) we obtain that (119906119905 120578

119905) isin

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) if 119903 lt infin and in

119871infin([0 119879] 119861

119904minus1

119901119903) times 119871

infin([0 119879] 119861

119904minus2

119901119903) otherwise Thus (119906 120578)

isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) Moreover a standard use of a sequence

of viscosity approximate solutions (119906120576 120578

120576)120576gt0

for (24) whichconverges uniformly in 119862([0 119879] 119861

119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) times

119862([0 119879] 119861119904minus1

119901119903) cap 119862

1([0 119879] 119861

119904minus2

119901119903) gives the continuity of

solution (119906 120578) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

32 Uniqueness and Continuity with Initial Data

Lemma 14 Assume that 1 le 119901 119903 le infin and 119904 gt max(1 +

1119901 32) Let (1199061 120578

1) and (119906

2 120578

2) be two given solutions to

the system (24) with initial data (1199061

0 120578

1

0) (119906

2

0 120578

2

0) isin 119861

119904

119901119903times

119861119904minus1

119901119903satisfying 119906

1 119906

2isin 119871

infin([0 119879] 119861

119904

119901119903) cap 119862([0 119879] 119861

119904minus1

119901119903) and

1205781 120578

2isin 119871

infin([0 119879] 119861

119904minus1

119901119903) cap 119862([0 119879] 119861

119904minus2

119901119903) Then for every

119905 isin [0 119879] one has

100381710038171003817100381710038171199061minus 119906

210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781minus 120578

210038171003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061

0minus 119906

2

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781

0minus 120578

2

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

times 119890

119862int119905

0[(1199061119861119904

119901119903

+1199062119861119904

119901119903

+119896+1)1199041+1

+1205781119861119904minus1

119901119903

+1205782119861119904minus1

119901119903

]d120591

(52)

Proof Denote 11990612

= 1199062minus 119906

1 120578

12= 120578

2minus 120578

1 then

11990612

isin 119871infin

([0 119879] 119861119904

119901119903) cap 119862 ([0 119879] 119861

119904minus1

119901119903)

12057812

isin 119871infin

([0 119879] 119861119904minus1

119901119903) cap 119862 ([0 119879] 119861

119904minus2

119901119903)

(53)

which implies that (11990612 120578

12) isin 119862([0 119879] 119861

119904minus1

119901119903) times 119862([0 119879]

119861119904minus2

119901119903) and (119906

12 120578

12) satisfies the following transport equation

12059711990511990612

+ 119906112059711990911990612

= minus119906121205971199091199062+ 119865

1(119905 119909)

119905 gt 0 119909 isin R

12059711990512057812

+ 119906112059711990912057812

= 1198650(119905 119909) 119905 gt 0 119909 isin R

11990612

(119909 0) = 11990612

0= 119906

2

0minus 119906

1

0 119909 isin R

12057812

(119909 0) = 12057812

0= 120578

2

0minus 120578

1

0 119909 isin R

(54)

with

1198651(119905 119909)

= 119875 (119863) [11990612

(1199061+ 119906

2) +

1

2

12059711990911990612120597119909(119906

1+ 119906

2)

+ 211989611990612

+

1

2

12057812

(1205781+ 120578

2+ 2)]

+ 1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

1198650(119905 119909) = minus119906

121205971199091205782minus (120578

121205971199091199061+ 120578

212059711990911990612

+ 12059711990911990612)

(55)

According to Lemma 12 we have

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585(

1003817100381710038171003817100381711990612120597119909119906210038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198651

1003817100381710038171003817B119904minus1119901119903

) d120591

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d12058510038171003817100381710038171198650

1003817100381710038171003817119861119904minus2

119901119903

d120591

(56)

Similar to the proof of the third step in Section 31 here weonly focus our attention on the high order terms We deducethat

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

(57)

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

for dissipative Camassa-Holm equation have been investi-gated For example Wu and Yin [19] studied the dissipativeCamassa-Holm equation

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120582 (119906 minus 119906

119909119909) = 2119906

119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

(3)

where 120582(119906 minus 119906119909119909

) (120582 gt 0) is the dissipative term Theyobtained the global existence result and blow-up results forstrong solutions in Sobolev space 119867

119904 with 119904 gt 32 byKatorsquos theory In [9] Lai andWu also investigated the weaklydissipative Camassa-Holm equation

119906119905minus 119906

119909119909119905+ 2119896119906

119909+ 3119906119906

119909+ 120582119906

2119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

(4)

where 119896 120582 120573 ge 0 are constants 119899119898 are natural numbersand 120582119906

2119899+1minus 120573119906

2119898119906119909119909

is the weakly dissipative term Theyobtained the local well-posedness in Sobolev space 119867

119904 with119904 gt 32 by using the pseudoparabolic regularization tech-nique and some estimates derived from the equation itselfand also developed a sufficient condition which guaranteedthe existence of weak solutions in Sobolev space 119867

119904 with1 lt 119904 le 32 We note that they only studied the equationin Sobolev space119867

119904On the other hand the Camassa-Holm equation also

admits many integrable multicomponent generalizations[20ndash32] For example

119898119905+ 2119896119906

119909+ 119906119898

119909+ 2119906

119909119898 + 120590120588120588

119909= 0

120588119905+ (120588119906)

119909= 0

(5)

where 119898 = 119906 minus 119906119909119909 The above system was derived in [33]

with 120590 = 1 In [20] Constantin and Ivanov gave a rigorousjustification of the system which is a valid approximation tothe governing equation for shallow water waves for 119896 = 0The 119906(119909 119905) represents the horizontal velocity of the fluid the120588(119909 119905) is related to the free surface elevation from equilibriumwith boundary assumptions and 119906 rarr 0 and 120588 rarr 1

as |119909| rarr infin They obtained the global well-posednesswith small initial data and the conditions for wave-breakingmechanism to the system For the case 120590 = minus1 it is notphysical as it corresponds to gravity pointing upwards and120588 rarr 0 as 119909 rarr infin The blow-up conditions are discussedin [25] For 119896 = 0 Gui and Liu [27] established the localwell-posedness for the system in a range of Besov spaces119861119904

119901119903times119861

119904minus1

119901119903with 119904 gt max(32 1+1119901)They also derivedwave-

breaking mechanisms and the exact blow-up rate for strongsolutions to the system in 119867

119904times 119867

119904minus1 with 119904 gt 32 Tian et al[34] obtained the local well-posedness for the system inBesov spaces 119861

119904

119901119903times 119861

119904

119901119903with 119904 gt max(32 1 + 1119901) They

also derived wave-breaking mechanisms for solutions anda result of blow-up solutions with certain profile in spaces119867119904times 119867

119904 with 119904 gt 32 Yan and Yin [21] investigated the 2-componentDegasperis-Procesi systemwhich is similar to the2-componentCamassa-Holm systemThey obtained the localwell-posedness in Besov spaces and derived a precise blow-up

scenario for strong solutions Guan and Yin [23] presented anew global existence result and several new blow-up resultsof strong solutions to an integrable 2-component Camassa-Holm shallow water system

In fact the 2-component Camassa-Holm system alsoadmits some generalizations due to the energy dissipationmechanisms in a real world In [35] Chen et al investigatedthe weakly dissipative 2-component Camassa-Holm system

119906119905minus 119906

119909119909119905+ 2119896119906

119909+ 3119906119906

119909minus 120588120588

119909+ 120582119906

2119899+1minus 120573119906

119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120588119905+ (120588119906)

119909= 0 119905 gt 0 119909 isin R

(6)

where 119896 120582 120573 ge 0 are constants and 119899 is a natural numberThey investigated the local well-posedness for the systemwithinitial data (119906

0 120588

0) isin 119867

119904times 119867

119904minus1 with 119904 ge 2 by Katorsquos theoryand derived a precise blow-up scenario for strong solutionsto the system

Motivated by the work in [9 21 23 27 33 35ndash38] westudy the weakly dissipative Camassa-Holm system (1) Wenote that the Cauchy problem of system (1) in Besov spaceshas not been discussed yet We state our main tasks withthree aspects Firstly we establish the local well-posednessof solutions to the system (1) Secondly we present theprecise blow-up criterions and exact blow-up rate for strongsolutions At last we derive a global existence result of strongsolutions Because of the presence of high order nonlinearterms 1199062119899+1 and 119906

2119898119906119909119909 the system (1) loses the conservation

law 119864 = intR(119906

2+ 119906

2

119909)d119909 which plays an important role in

studying system (1)Now we rewrite system (1) as

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1

2

1205882]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

2119898119906119909119909

]

119905 gt 0 119909 isin R

120588119905+ (120588119906)

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120588 (119909 0) = 1205880(119909) 119909 isin R

(7)

where the operators 119875(119863) = minus120597119909(1 minus 120597

119909119909)minus1 and 119875

1(119863) = (1 minus

120597119909119909

)minus1 We define the space

119864119904

119901119903(119879)

=

119862([0 119879] 119861119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) 1 le 119903 lt infin

119871infin

([0 119879] 119861119904

119901infin) cap lip ([0 119879] 119861

119904minus1

119901infin) 119903 = infin

(8)

with 119879 gt 0 119904 isin R 119901 isin [1infin] and 119903 isin [1infin]Themain results of this paper are stated as follows Firstly

we present the local well-posedness theorem

Mathematical Problems in Engineering 3

Theorem 1 For 1 le 119901 119903 le infin and 119904 gt max(32 1 + 1119901)Let (119906

0 120588

0minus 1) isin 119861

119904

119901119903times 119861

119904minus1

119901119903 There exists a time 119879 gt 0 such

that the initial value problem (1) has a unique solution (119906 120588 minus

1) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) and the map (119906

0 120588

0minus 1) rarr (119906 120588 minus

1) is continuous from a neighborhood of (1199060 120588

0minus 1) in 119861

119904

119901119903times

119861119904minus1

119901119903into 119862([0 119879] 119861

1199041015840

119901119903) cap 119862

1([0 119879] 119861

1199041015840minus1

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) cap

1198621([0 119879] 119861

1199041015840minus2

119901119903) for every 119904

1015840lt 119904 when 119903 = +infin and 119904

1015840= 119904

whereas 119903 lt +infin

We obtain the following blow-up results

Theorem 2 Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with 119904 gt 52 and119879 gt 0 be the maximal existence time of the solution (119906 120588) tosystem (1) with initial data (119906

0 120588

0) Then the corresponding

solution blows up in finite time if and only if

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909) = minusinfin

119900119903

lim119905rarr119879

minus

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin

(9)

Theorem 3 Let 119898 = 0 in system (1) Assume (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the initial value (1199060 120588

0) satisfies

1199060119909(119909

0) lt minus120573 minus radic2119870 where 119870 is a fixed constant defined in

(95) and 1205880(119909

0) = 0 with the point 119909

0defined by 119906

0119909(119909

0) =

inf119909isinR1199060119909(119909) Then the corresponding solution (119906 120588 minus 1) to

system (1) with 119898 = 0 blows up in finite time Namely thereexists a 119879

0with 0 lt 119879

0le minus2(1 minus 120575)(119906

0119909(119909

0) + 120573) such that

lim inf119905rarr119879

minus

0

inf119909isinR

119906119909(119905 119909) = minusinfin (10)

where 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870

Theorem 4 Let 119896 = 0 in system (1) Assume (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and the initial value (1199060 120588

0) satisfies

that 1199060is odd 120588

0is even and 119906

0119909(0) lt minusradic2119870

1 where 119870

1

is a fixed constant defined in (109) and 1205880(0) = 0 Then the

corresponding solution (119906 120588 minus 1) to system (1) with 119896 = 0

blows up in finite time More precisely there exists a 1198791with

0 lt 1198791le minus2(1 minus 120575

1)119906

0119909(0) such that

lim inf119905rarr119879

minus

1

119906119909(119905 0) = minusinfin (11)

In addition if 1205880119909(119909

0) = 0 with some 119909

0isin R satisfying

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) then there exists a 1198792with 0 lt 119879

2le

minus(2((1 minus 1205751)119906

0119909(0))) such that

(i) lim suptrarrTminus2

supxisinR

120588x (t x) = +infin if 1205880x (x0) gt 0

(ii) lim inf119905rarr119879

minus

2

inf119909isinR

120588119909(119905 119909) = minusinfin 119894119891 120588

0119909(119909

0) lt 0

(12)

where 1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1

We also obtain the exact blow-up rate of strong solutionsto system (1)

Theorem 5 Let 119879 lt infin be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) with119898 = 0 Theinitial data (119906

0 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with 119904 gt 32 satisfies1199060119909(119909

0) lt minus120573 minus radic2119870

2 where 119870

2is a fixed constant defined

in (120) and 1205880(119909

0) = 0 with the point 119909

0defined by 119906

0(119909

0) =

inf119909isinR1199060119909(119909) then

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (13)

Now we present a global existence result of strongsolutions to system (1)

Theorem6 Let119898 = 0 in system (1) and (1199060 120588

0minus1) isin 119867

2times119867

1If120588

0(119909) = 0 for all119909 isin R then the corresponding strong solution

(119906 120588 minus 1) to system (1) with 119898 = 0 exists globally in time

The remainder of this paper is organized as followsIn Section 2 some properties of Besov space and a prioriestimates for solutions of transport equation are reviewedSection 3 is devoted to the proof of Theorem 1 The proofs ofwave-breaking results and the precise blow-up rate of strongsolutions to system (1) are given in Section 4 respectivelyTheproof of Theorem 6 is presented in Section 5

Notation In this paper we denote the convolution onR by lowast

the norm of Lebesgue space 119871119901 by sdot

119871119901 1 le 119901 le infin and the

norm in Sobolev space 119867119904 119904 isin R by sdot

119867119904 and the norm in

Besov space119861119904119901119903

119904 isin R by sdot119861119904

119901119903

Here we denote 119886+ = 119886+120576where 120576 gt 0 is a sufficiently small number

2 Preliminary

In this section we recall some basic facts in Besov space Onemay check [21 39ndash42] for more details

Proposition 7 (see [40 42]) Let 119904 isin R 1 le 119901 and 119903 le +infinThe nonhomogeneous Besov space is defined by 119861

119904

119901119903= 119891 isin

1198781015840(R119899

) | 119891119861119904

119901119903

lt +infin where

10038171003817100381710038171198911003817100381710038171003817119861119904

119901119903

=

(

infin

sum

119895=minus1

211989511990311990410038171003817

100381710038171003817Δ119895119891

10038171003817100381710038171003817

119903

119871p)

1119903

119903 lt infin

sup119895geminus1

211989511990410038171003817100381710038171003817Δ119895119891

10038171003817100381710038171003817119871119901 119903 = infin

(14)

Moreover 119878119895119891 = sum

119895minus1

119902=minus1Δ119902119891

Proposition 8 (see [40 42]) Let 119904 isin R 1 le 119901 119903 119901119895 119903119895le infin

and 119895 = 1 2 then consider the following

(1) Embedding 11986111990411990111199031

997893rarr 119861119904minus119899(1119901

1minus11199012)

11990121199032

if 1199011

le 1199012and

1199031le 119903

2 And 119861

1199042

1199011199032

997893rarr 1198611199041

1199011199031

locally compact if 1199041le 119904

2

(2) Algebraic properties for any 119904 gt 0 119861119904

119901119903cap 119871

infin is analgebra 119861119904

119901119903is an algebra hArr 119861

119904

119901119903997893rarr 119871

infinhArr 119904 gt

119899119901 119900119903 119904 ge 119899119901 119886119899119889 119903 = 1

4 Mathematical Problems in Engineering

(3) Complex interpolation

100381710038171003817100381711989110038171003817100381710038171198611205791199041+(1minus120579)1199042

119901119903

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

1198611199041

119901119903

10038171003817100381710038171198911003817100381710038171003817

1minus120579

1198611199042

119901119903

119891119900119903 119890V119890119903119910 119891 isin 1198611199041

119901119903cap 119861

1199042

119901119903 120579 isin [0 1]

(15)

(4) Fatoursquos Lemma if (119906119899)119899isinN is bounded in119861

119904

119901119903and 119906

119899rarr

119906 in 1198781015840(R) then 119906 isin 119861

119904

119901119903and

119906119861119904

119901119903

le lim119899rarrinfin

inf 1003817100381710038171003817119906119899

1003817100381710038171003817119861119904

119901119903

(16)

(5) Let 119898 isin R and 119891 be an 119878119898-multiplier (ie 119891 R119899

rarr

R is smooth and satisfies that for every 120572 isin N119899 thereexists a constant 119862

120572such that |120597

120572119891(120585)| le 119862

120572(1 +

|120585|)119898minus|120572| for all 120585 isin R119899) Then the operator 119891(119863) is

continuous from 119861119904

119901119903to 119861

119904minus119898

119901119903

(6) Density 119862infin

119888is dense in 119861

119904

119901119903hArr 1 le 119901 119903 lt infin

Lemma 9 (see [40]) Let 119868 be an open interval of R Let 119904 gt 0

and 1199041be the smallest integer such that 119904

1ge 119904 Let 119865 119868 rarr R

satisfy 119865(0) = 0 and 1198651015840isin 119882

1199041infin

(119868R) Assume that V isin 119861119904

119901119903

has values in 119869 subsub 119868 Then 119865(V) isin 119861119904

119901119903and there exists a

constant 119862 depending only on 119904 119868 119869119873 such that

119865 (V)119861119904

119901119903

le 119862(1 + V119871infin)

1199041100381710038171003817100381710038171198651015840100381710038171003817100381710038171198821199041infin(119868)

V119861119904

119901119903

(17)

Lemma 10 (see [40]) Let 119868 be an open interval of R Let119904 gt 0 and 119904

1be the smallest integer such that 119904

1ge 119904 Let

119865 119868 rarr R satisfy 1198651015840(0) = 0 and 119865

10158401015840isin 119882

1199041infin

(119868R) Assumethat 119906 V isin 119861

119904

119901119903has values in 119869 subsub 119868 Then there exists a

constant 119862 depending only on 119904 119868 119869119873 such that

119865 (119906) minus 119865 (V)119861119904

119901119903

le 119862(1 + 119906119871infin)

11990411003817100381710038171003817100381711986510158401015840100381710038171003817100381710038171198821199041infin(119868)

times (119906 minus V119861119904

119901119903

sup120579isin[01]

120579119906 + (1 minus 120579) V119871infin

+ 119906 minus V119871infin sup120579isin[01]

120579119906 + (1 minus 120579) V119861119904

119901119903

)

(18)

Lemma 11 (see [40]) Assume that 1 le 119901 and 119903 le infin thefollowing estimates hold

(i) for 119904 gt 0 then

1003817100381710038171003817119891119892

1003817100381710038171003817119861119904

119901119903

le 119862(10038171003817100381710038171198911003817100381710038171003817119861119904

119901119903

10038171003817100381710038171198921003817100381710038171003817119871infin +

10038171003817100381710038171198911003817100381710038171003817119871infin

10038171003817100381710038171198921003817100381710038171003817119861119904

119901119903

)

(19)

(ii) for 1199041

le 1119901 1199042

gt 1119901 (1199042

ge (1119901) if 119903 = 1) and1199041+ 119904

2gt 0 then

1003817100381710038171003817119891119892

10038171003817100381710038171198611199041

119901119903

le 119862100381710038171003817100381711989110038171003817100381710038171198611199041

119901119903

100381710038171003817100381711989210038171003817100381710038171198611199042

119901119903

(20)

where 119862 are constants independent of 119891 119892

Then we present two related lemmas for the followingtransport equations

119891119905+ 119889 sdot nabla119891 = 119865

1198911003816100381610038161003816119905=0

= 1198910

(21)

where 119889 R times R119899rarr R119899 stands for a given time dependent

vector field and 1198910 R119899

rarr R119898 and 119865 R times R119899rarr R119898 are

known data

Lemma 12 (see [39 40]) Let 1 le 119901 le 1199011le infin 1 le 119903 le infin

and 1199011015840= 119901(119901 minus 1) Assume that 119904 gt minus119899 sdot min(1119901

1 1119901

1015840)

or 119904 gt minus1 minus 119899 sdot min(11199011 1119901

1015840) if nabla sdot 119889 = 0 Then there

exists a constant 119862 depending only on 119899 119901 1199011 119903 119904 such that

the following estimate holds true

10038171003817100381710038171198911003817100381710038171003817infin

119905([0119905]119861

119904

119901119903)

le 1198901198621int119905

0119885(120591)d120591

times [10038171003817100381710038171198910

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890minus1198621int120591

0119885(120585)119889120585

119865 (120591)119861119904

119901119903

d120591]

(22)

with

119885 (119905)

=

nabla119889 (119905)1198611198991199011

1199011infincap119871infin 119904 lt 1 +

119899

1199011

nabla119889 (119905)119861119904minus1

1199011119903

119904 gt 1+

119899

1199011

119900119903 119904=1+

119899

1199011

119903=1

(23)

If 119891 = 119889 then for all 119904 gt 0 (nabla sdot 119889 = 0 119904 gt minus1) (22) holds with119885(119905) = nabla119889(119905)

119871infin

Let us state the existence result for transport equationwith data in Besov space

Lemma 13 (see [40]) Let 119901 1199011 119903 119904 be as in the statement

of Lemma 12 and 1198910

isin 119861119904

119901119903and 119865 isin 119871

1([0 119879] 119861

119904

119901119903) 119889 isin

119871120588([0 119879] 119861

minus119872

infininfin) is a time dependent vector field for some

120588 gt 1119872 gt 0 such that if 119904 lt 1 + 1198991199011then nabla119889 isin

1198711([0 119879] 119861

1198991199011

1199011infin

cap 119871infin) if 119904 gt 1 + 119899119901

1or 119904 = 1 + 119899119901

1 119903 = 1

then nabla119889 isin 1198711([0 119879] 119861

119904minus1

1199011119903) Thus transport equation (21) has a

unique solution 119891 isin 119871infin([0 119879] 119861

119904

119901119903)⋂(⋂

1199041015840lt119904

119862([0 119879] 1198611199041015840

1199011))

and (22) holds true If 119903 lt infin then one has 119891 isin 119862([0 119879] 119861119904

119901119903)

3 The Proof of Theorem 1

We finish the proof of Theorem 1 by the following steps

Mathematical Problems in Engineering 5

31 Existence of Solutions For convenience we denote 120578 =

120588 minus 1 and rewrite (7) as

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205782

2

+ 120578]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

2119898119906119909119909

]

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909= minus119906

119909120578 minus 119906

119909 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(24)

We use a standard iterative process to construct the approxi-mate solutions to (24)

Step 1 Starting from 1199060

= 1205780

= 0 we define by inductiona sequence of smooth functions (119906

119894 120578

119894)119894isinN isin 119862(R+

119861infin

119901119903)2

solving the following transport equation

(120597119905+ 119906

119894120597119909) 119906

119894+1= 119865 (119905 119909) 119905 gt 0 119909 isin R

(120597119905+ 119906

119894120597119909) 120578

119894+1= minus120578

119894120597119909119906119894minus 120597

119909119906119894 119905 gt 0 119909 isin R

119906119894+1

(119909 0) = 119906119894+1

0(119909) = 119878

119894+11199060 119909 isin R

120578119894+1

(119909 0) = 120578119894+1

0(119909) = 119878

119894+11205780 119909 isin R

(25)

where

119865 (119905 119909) = 119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+ 120578119894]

+ 1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

(26)

Since all the data 119878119894+1

1199060 119878

119894+11205780isin 119861

infin

119901119903 Lemma 13 enables us

to show that for all 119894 isin N the system (25) has a global solutionwhich belongs to 119862(R+

119861infin

119901119903)2

Step 2 Next we prove that (119906119894 120578119894)119894isinN is uniformly bounded

in 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

According to Lemma 12 we have the following inequali-ties for all 119894 isin N10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

le 119890

1198621int119905

0119906119894(120591)119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890

minus1198621int120591

0119906119894(120585)119861119904

119901119903

d120585119865 (120591 sdot)

119861119904

119901119903

d120591]

(27)

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

1198622int119905

0119906119894119861119904

119901119903

d120591[10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198622int120591

0119906119894119861119904

119901119903

d120585

times

10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

d120591]

(28)

From Proposition 8 Lemma 11 and 119878minus1-multiplier property

of 119875(119863) it yields that1003817100381710038171003817100381710038171003817

119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+120578119894]

1003817100381710038171003817100381710038171003817119861119904

119901119903

le 119862(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

119861119904

119901119903

+2119896

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817

2

119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

(29)

Using the 119878minus2-multiplier property of 119875

1(119863) we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862

100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

(30)

(i) By the assumption of 119891(119906119894) = (119906

119894)2119899+1 and Lemma 9

we have100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119871infin)

(119904minus1)110038171003817100381710038171003817119891101584010038171003817100381710038171003817119882(119904minus1)1infin

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

(119904minus1)11003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(31)

(ii) As we know if max(32 1 + 1119901) lt 119904 le 2 + 1119901 then119861119904minus1

119901119903is an algebra if 119904 gt 2+1119901 then119861

119904minus2

119901119903is an algebra

By the assumption of ℎ(119906119894) = (119906119894)2119898 and Lemmas 10

and 11 we deduce that100381710038171003817100381710038171003817

120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862120573

100381710038171003817100381710038171003817

(119906119894)

2119898100381710038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

(119904minus1)1

times

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(32)

6 Mathematical Problems in Engineering

Combining (i) and (ii) yields100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(33)

Thanks to Lemma 11 we get10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(34)

Therefore from (27) to (34) we obtain10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 1198623sdot 119890

1198623int119905

0119906119894119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198623int120591

0119906119894119861119904

119901119903

d120585(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 119896 + 1)

1199041

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)] d120591

(35)

Let us choose a 119879 gt 0 such that 211990411198621199041+1

3( 119906

119894

0119861119904

119901119903

+ 120578119894

0119861119904minus1

119901119903

+

119896 + 1)1199041119879 lt 1 and

(119896 + 1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(36)

Inserting (36) into (35) yields

(119896 + 1 +

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(37)

Therefore (119906119894 120578119894)119894isinN is uniformly bounded in119862([0 119879] 119861

119904

119901119903)times

119862([0 119879] 119861119904minus1

119901119903) From Proposition 8 Lemma 11 and the fol-

lowing embedding properties

119861119904

119901119903997893rarr 119861

119904minus1

119901119903 119861

119904minus1

119901119903997893rarr 119861

119904minus2

119901119903 (38)

we have10038171003817100381710038171003817119906119894120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

(39)

We deduce that 119906119894120597119909119906119894+1 and 119865(119905 119909) are uniformly bounded

in 119862([0 119879] 119861119904minus1

119901119903) in the same way that 119906

119894120597119909120578119894+1

minus120578119894120597119909119906119894minus

120597119909119906119894 are uniformly bounded in 119862([0 119879] 119861

119904minus2

119901119903) Using the

system (25) we have (120597119905119906119894+1

120597119905120578119894+1

) isin 119862([0 119879] 119861119904minus1

119901119903) times

119862([0 119879] 119861119904minus2

119901119903) is uniformly bounded which derives that

(119906119894 120578

119894)119894isinN is uniformly bounded in 119864

119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

Step 3 Now we demonstrate that (119906119894 120578

119894)119894isinN is a Cauchy

sequence in 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

In fact according to (25) we note that for all 119894 119895 isin N wehave

(120597119905+ 119906

119894+119895120597119909) (119906

119894+119895+1minus 119906

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909119906119894+1

+ 119875 (119863)

times [(119906119894+119895

minus 119906119894) (119906

119894+119895+ 119906

119894) +

1

2

120597119909(119906

119894+119895minus 119906

119894)

sdot 120597119909(119906

119894+119895+ 119906

119894) + 2119896 (119906

119894+119895minus 119906

119894) +

1

2

(120578119894+119895

minus 120578119894)

times (120578119894+119895

+ 120578119894) + (120578

119894+119895minus 120578

119894) ]

+ 1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)

+ 120573 ((119906119894+119895

)

2119898

(119906119894+119895

)119909119909

minus (119906119894)

2119898

(119906119894)119909119909

)]

(40)

(120597119905+ 119906

119894+119895120597119909) (120578

119894+119895+1minus 120578

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909120578119894+1

minus (120578119894+119895

minus 120578119894) 120597

119909119906119894

minus 120578119894+119895

120597119909(119906

119894+119895minus 119906

119894) minus 120597

119909(119906

119894+119895minus 119906

119894)

(41)

(1) We estimate the terms in the right side of (40) ByLemma 11 we have10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (119906

119894+119895minus 119906

119894) (119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894+119895

+ 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [120597

119909(119906

119894+119895minus 119906

119894) 120597

119909(119906

119894+119895+ 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [2119896 (119906

119894+119895minus 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862119896

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817119875 (119863) [(120578

119894+119895minus 120578

119894) (120578

119894+119895+ 120578

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+119895

+ 12057811989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (120578

119894+119895minus 120578

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

(42)

Mathematical Problems in Engineering 7

We use Lemma 10 to estimate the high order terms in (40)Firstly we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

119904110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(43)

Secondly by 1199062119898

119906119909119909

= 120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909and

Lemma 11 we deduce that100381710038171003817100381710038171003817

120573 ((119906119894+119895

)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

le 119862(

100381710038171003817100381710038171003817

120597119909((119906

119894+119895)

2119898

119906119894+119895

119909minus (119906

119894)

2119898

119906119894

119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

+

100381710038171003817100381710038171003817

(119906119894+119895

)

2119898minus1

(119906119894+119895

119909)

2

minus (119906119894)

2119898minus1

(119906119894

119909)

2100381710038171003817100381710038171003817119861119904minus3

119901119903

)

(44)

Hence100381710038171003817100381710038171003817

1198751(119863) 120573 [(119906

119894+119895)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

1199041+110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(45)

(2) Now we estimate the right side of (41)10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817(120578

119894+119895minus 120578

119894) 120597

11990911990611989410038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

1003817100381710038171003817100381712059711990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120578119894+119895

120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(46)

Combining (1) (2) and Lemma 12 yields for all 119905 isin [0 119879]

10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817119906119894+119895+1

0minus 119906

119894+1

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862

times int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)(

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 1)

+

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+ 119896 + 1)

1199041+1

] d120591

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817120578119894+119895+1

0minus 120578

119894+1

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+ 1)

+(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

] d120591

(47)

Since (119906119894 120578119894)119894isinN is uniformly bounded in119864

119904

119901119903(119879)times119864

119904minus1

119901119903(119879) and

119906119894+119895+1

0minus 119906

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021199060

120578119894+119895+1

0minus 120578

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021205780

(48)

there exists a constant 119862119879independent of 119894 119895 such that for all

119905 isin [0 119879] we get10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862119879[2

minus119894+ int

119905

0

(

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

) d120591]

(49)

By induction we obtain that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le

(119862119879119879)

119894+1

(119894 + 1)

[

1003817100381710038171003817100381711990611989510038171003817100381710038171003817119871infin([0119879]119861

119904minus1

119901119903)+

1003817100381710038171003817100381712057811989510038171003817100381710038171003817119871infin([0119879]119861

119904minus2

119901119903)]

+ 119862119879

119894

sum

119897=0

2minus(119894minus119897)

(119862119879119879)

119897

119897

(50)

Since 119906119895119871infin

119879(119861119904

119901119903) 120578

119895119871infin

119879(119861119904minus1

119901119903)are bounded independently of

119895 we conclude that there exists a new constant 1198621198791

such that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le 1198621198791

2minus119894

(51)

Consequently (119906119894 120578119894)119894isinN is a Cauchy sequence in 119862([0 119879]

119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

8 Mathematical Problems in Engineering

Step 4 We end the proof of existence of solutions

Firstly since (119906119894 120578

119894)119894isinN is uniformly bounded in

119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) according to the Fatou

property for Besov space it guarantees that (119906 120578) alsobelongs to 119871

infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903)

Secondly for (119906119894 120578

119894)119894isinN is a Cauchy sequence in

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) so it converges to some

limit function (119906 120578) isin 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) An

interpolation argument insures that the convergence holdsin 119862([0 119879] 119861

1199041015840

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904 It is easy to

pass to the limit in (25) and to conclude that (119906 120578) is indeeda solution to (24) Thanks to the fact that (119906 120578) belongs to119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) we know that the right

side of the first equation in (24) belongs to 119871infin([0 119879] 119861

119904

119901119903)

and the right side of the second equation in (24) belongs to119871infin([0 119879] 119861

119904minus1

119901119903) In the case of 119903 lt infin applying Lemma 13

derives (119906 120578) isin 119862([0 119879] 1198611199041015840

119901119903) times119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904

Finally with (24) we obtain that (119906119905 120578

119905) isin

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) if 119903 lt infin and in

119871infin([0 119879] 119861

119904minus1

119901119903) times 119871

infin([0 119879] 119861

119904minus2

119901119903) otherwise Thus (119906 120578)

isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) Moreover a standard use of a sequence

of viscosity approximate solutions (119906120576 120578

120576)120576gt0

for (24) whichconverges uniformly in 119862([0 119879] 119861

119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) times

119862([0 119879] 119861119904minus1

119901119903) cap 119862

1([0 119879] 119861

119904minus2

119901119903) gives the continuity of

solution (119906 120578) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

32 Uniqueness and Continuity with Initial Data

Lemma 14 Assume that 1 le 119901 119903 le infin and 119904 gt max(1 +

1119901 32) Let (1199061 120578

1) and (119906

2 120578

2) be two given solutions to

the system (24) with initial data (1199061

0 120578

1

0) (119906

2

0 120578

2

0) isin 119861

119904

119901119903times

119861119904minus1

119901119903satisfying 119906

1 119906

2isin 119871

infin([0 119879] 119861

119904

119901119903) cap 119862([0 119879] 119861

119904minus1

119901119903) and

1205781 120578

2isin 119871

infin([0 119879] 119861

119904minus1

119901119903) cap 119862([0 119879] 119861

119904minus2

119901119903) Then for every

119905 isin [0 119879] one has

100381710038171003817100381710038171199061minus 119906

210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781minus 120578

210038171003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061

0minus 119906

2

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781

0minus 120578

2

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

times 119890

119862int119905

0[(1199061119861119904

119901119903

+1199062119861119904

119901119903

+119896+1)1199041+1

+1205781119861119904minus1

119901119903

+1205782119861119904minus1

119901119903

]d120591

(52)

Proof Denote 11990612

= 1199062minus 119906

1 120578

12= 120578

2minus 120578

1 then

11990612

isin 119871infin

([0 119879] 119861119904

119901119903) cap 119862 ([0 119879] 119861

119904minus1

119901119903)

12057812

isin 119871infin

([0 119879] 119861119904minus1

119901119903) cap 119862 ([0 119879] 119861

119904minus2

119901119903)

(53)

which implies that (11990612 120578

12) isin 119862([0 119879] 119861

119904minus1

119901119903) times 119862([0 119879]

119861119904minus2

119901119903) and (119906

12 120578

12) satisfies the following transport equation

12059711990511990612

+ 119906112059711990911990612

= minus119906121205971199091199062+ 119865

1(119905 119909)

119905 gt 0 119909 isin R

12059711990512057812

+ 119906112059711990912057812

= 1198650(119905 119909) 119905 gt 0 119909 isin R

11990612

(119909 0) = 11990612

0= 119906

2

0minus 119906

1

0 119909 isin R

12057812

(119909 0) = 12057812

0= 120578

2

0minus 120578

1

0 119909 isin R

(54)

with

1198651(119905 119909)

= 119875 (119863) [11990612

(1199061+ 119906

2) +

1

2

12059711990911990612120597119909(119906

1+ 119906

2)

+ 211989611990612

+

1

2

12057812

(1205781+ 120578

2+ 2)]

+ 1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

1198650(119905 119909) = minus119906

121205971199091205782minus (120578

121205971199091199061+ 120578

212059711990911990612

+ 12059711990911990612)

(55)

According to Lemma 12 we have

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585(

1003817100381710038171003817100381711990612120597119909119906210038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198651

1003817100381710038171003817B119904minus1119901119903

) d120591

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d12058510038171003817100381710038171198650

1003817100381710038171003817119861119904minus2

119901119903

d120591

(56)

Similar to the proof of the third step in Section 31 here weonly focus our attention on the high order terms We deducethat

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

(57)

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Theorem 1 For 1 le 119901 119903 le infin and 119904 gt max(32 1 + 1119901)Let (119906

0 120588

0minus 1) isin 119861

119904

119901119903times 119861

119904minus1

119901119903 There exists a time 119879 gt 0 such

that the initial value problem (1) has a unique solution (119906 120588 minus

1) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) and the map (119906

0 120588

0minus 1) rarr (119906 120588 minus

1) is continuous from a neighborhood of (1199060 120588

0minus 1) in 119861

119904

119901119903times

119861119904minus1

119901119903into 119862([0 119879] 119861

1199041015840

119901119903) cap 119862

1([0 119879] 119861

1199041015840minus1

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) cap

1198621([0 119879] 119861

1199041015840minus2

119901119903) for every 119904

1015840lt 119904 when 119903 = +infin and 119904

1015840= 119904

whereas 119903 lt +infin

We obtain the following blow-up results

Theorem 2 Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with 119904 gt 52 and119879 gt 0 be the maximal existence time of the solution (119906 120588) tosystem (1) with initial data (119906

0 120588

0) Then the corresponding

solution blows up in finite time if and only if

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909) = minusinfin

119900119903

lim119905rarr119879

minus

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin

(9)

Theorem 3 Let 119898 = 0 in system (1) Assume (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the initial value (1199060 120588

0) satisfies

1199060119909(119909

0) lt minus120573 minus radic2119870 where 119870 is a fixed constant defined in

(95) and 1205880(119909

0) = 0 with the point 119909

0defined by 119906

0119909(119909

0) =

inf119909isinR1199060119909(119909) Then the corresponding solution (119906 120588 minus 1) to

system (1) with 119898 = 0 blows up in finite time Namely thereexists a 119879

0with 0 lt 119879

0le minus2(1 minus 120575)(119906

0119909(119909

0) + 120573) such that

lim inf119905rarr119879

minus

0

inf119909isinR

119906119909(119905 119909) = minusinfin (10)

where 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870

Theorem 4 Let 119896 = 0 in system (1) Assume (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and the initial value (1199060 120588

0) satisfies

that 1199060is odd 120588

0is even and 119906

0119909(0) lt minusradic2119870

1 where 119870

1

is a fixed constant defined in (109) and 1205880(0) = 0 Then the

corresponding solution (119906 120588 minus 1) to system (1) with 119896 = 0

blows up in finite time More precisely there exists a 1198791with

0 lt 1198791le minus2(1 minus 120575

1)119906

0119909(0) such that

lim inf119905rarr119879

minus

1

119906119909(119905 0) = minusinfin (11)

In addition if 1205880119909(119909

0) = 0 with some 119909

0isin R satisfying

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) then there exists a 1198792with 0 lt 119879

2le

minus(2((1 minus 1205751)119906

0119909(0))) such that

(i) lim suptrarrTminus2

supxisinR

120588x (t x) = +infin if 1205880x (x0) gt 0

(ii) lim inf119905rarr119879

minus

2

inf119909isinR

120588119909(119905 119909) = minusinfin 119894119891 120588

0119909(119909

0) lt 0

(12)

where 1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1

We also obtain the exact blow-up rate of strong solutionsto system (1)

Theorem 5 Let 119879 lt infin be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) with119898 = 0 Theinitial data (119906

0 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with 119904 gt 32 satisfies1199060119909(119909

0) lt minus120573 minus radic2119870

2 where 119870

2is a fixed constant defined

in (120) and 1205880(119909

0) = 0 with the point 119909

0defined by 119906

0(119909

0) =

inf119909isinR1199060119909(119909) then

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (13)

Now we present a global existence result of strongsolutions to system (1)

Theorem6 Let119898 = 0 in system (1) and (1199060 120588

0minus1) isin 119867

2times119867

1If120588

0(119909) = 0 for all119909 isin R then the corresponding strong solution

(119906 120588 minus 1) to system (1) with 119898 = 0 exists globally in time

The remainder of this paper is organized as followsIn Section 2 some properties of Besov space and a prioriestimates for solutions of transport equation are reviewedSection 3 is devoted to the proof of Theorem 1 The proofs ofwave-breaking results and the precise blow-up rate of strongsolutions to system (1) are given in Section 4 respectivelyTheproof of Theorem 6 is presented in Section 5

Notation In this paper we denote the convolution onR by lowast

the norm of Lebesgue space 119871119901 by sdot

119871119901 1 le 119901 le infin and the

norm in Sobolev space 119867119904 119904 isin R by sdot

119867119904 and the norm in

Besov space119861119904119901119903

119904 isin R by sdot119861119904

119901119903

Here we denote 119886+ = 119886+120576where 120576 gt 0 is a sufficiently small number

2 Preliminary

In this section we recall some basic facts in Besov space Onemay check [21 39ndash42] for more details

Proposition 7 (see [40 42]) Let 119904 isin R 1 le 119901 and 119903 le +infinThe nonhomogeneous Besov space is defined by 119861

119904

119901119903= 119891 isin

1198781015840(R119899

) | 119891119861119904

119901119903

lt +infin where

10038171003817100381710038171198911003817100381710038171003817119861119904

119901119903

=

(

infin

sum

119895=minus1

211989511990311990410038171003817

100381710038171003817Δ119895119891

10038171003817100381710038171003817

119903

119871p)

1119903

119903 lt infin

sup119895geminus1

211989511990410038171003817100381710038171003817Δ119895119891

10038171003817100381710038171003817119871119901 119903 = infin

(14)

Moreover 119878119895119891 = sum

119895minus1

119902=minus1Δ119902119891

Proposition 8 (see [40 42]) Let 119904 isin R 1 le 119901 119903 119901119895 119903119895le infin

and 119895 = 1 2 then consider the following

(1) Embedding 11986111990411990111199031

997893rarr 119861119904minus119899(1119901

1minus11199012)

11990121199032

if 1199011

le 1199012and

1199031le 119903

2 And 119861

1199042

1199011199032

997893rarr 1198611199041

1199011199031

locally compact if 1199041le 119904

2

(2) Algebraic properties for any 119904 gt 0 119861119904

119901119903cap 119871

infin is analgebra 119861119904

119901119903is an algebra hArr 119861

119904

119901119903997893rarr 119871

infinhArr 119904 gt

119899119901 119900119903 119904 ge 119899119901 119886119899119889 119903 = 1

4 Mathematical Problems in Engineering

(3) Complex interpolation

100381710038171003817100381711989110038171003817100381710038171198611205791199041+(1minus120579)1199042

119901119903

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

1198611199041

119901119903

10038171003817100381710038171198911003817100381710038171003817

1minus120579

1198611199042

119901119903

119891119900119903 119890V119890119903119910 119891 isin 1198611199041

119901119903cap 119861

1199042

119901119903 120579 isin [0 1]

(15)

(4) Fatoursquos Lemma if (119906119899)119899isinN is bounded in119861

119904

119901119903and 119906

119899rarr

119906 in 1198781015840(R) then 119906 isin 119861

119904

119901119903and

119906119861119904

119901119903

le lim119899rarrinfin

inf 1003817100381710038171003817119906119899

1003817100381710038171003817119861119904

119901119903

(16)

(5) Let 119898 isin R and 119891 be an 119878119898-multiplier (ie 119891 R119899

rarr

R is smooth and satisfies that for every 120572 isin N119899 thereexists a constant 119862

120572such that |120597

120572119891(120585)| le 119862

120572(1 +

|120585|)119898minus|120572| for all 120585 isin R119899) Then the operator 119891(119863) is

continuous from 119861119904

119901119903to 119861

119904minus119898

119901119903

(6) Density 119862infin

119888is dense in 119861

119904

119901119903hArr 1 le 119901 119903 lt infin

Lemma 9 (see [40]) Let 119868 be an open interval of R Let 119904 gt 0

and 1199041be the smallest integer such that 119904

1ge 119904 Let 119865 119868 rarr R

satisfy 119865(0) = 0 and 1198651015840isin 119882

1199041infin

(119868R) Assume that V isin 119861119904

119901119903

has values in 119869 subsub 119868 Then 119865(V) isin 119861119904

119901119903and there exists a

constant 119862 depending only on 119904 119868 119869119873 such that

119865 (V)119861119904

119901119903

le 119862(1 + V119871infin)

1199041100381710038171003817100381710038171198651015840100381710038171003817100381710038171198821199041infin(119868)

V119861119904

119901119903

(17)

Lemma 10 (see [40]) Let 119868 be an open interval of R Let119904 gt 0 and 119904

1be the smallest integer such that 119904

1ge 119904 Let

119865 119868 rarr R satisfy 1198651015840(0) = 0 and 119865

10158401015840isin 119882

1199041infin

(119868R) Assumethat 119906 V isin 119861

119904

119901119903has values in 119869 subsub 119868 Then there exists a

constant 119862 depending only on 119904 119868 119869119873 such that

119865 (119906) minus 119865 (V)119861119904

119901119903

le 119862(1 + 119906119871infin)

11990411003817100381710038171003817100381711986510158401015840100381710038171003817100381710038171198821199041infin(119868)

times (119906 minus V119861119904

119901119903

sup120579isin[01]

120579119906 + (1 minus 120579) V119871infin

+ 119906 minus V119871infin sup120579isin[01]

120579119906 + (1 minus 120579) V119861119904

119901119903

)

(18)

Lemma 11 (see [40]) Assume that 1 le 119901 and 119903 le infin thefollowing estimates hold

(i) for 119904 gt 0 then

1003817100381710038171003817119891119892

1003817100381710038171003817119861119904

119901119903

le 119862(10038171003817100381710038171198911003817100381710038171003817119861119904

119901119903

10038171003817100381710038171198921003817100381710038171003817119871infin +

10038171003817100381710038171198911003817100381710038171003817119871infin

10038171003817100381710038171198921003817100381710038171003817119861119904

119901119903

)

(19)

(ii) for 1199041

le 1119901 1199042

gt 1119901 (1199042

ge (1119901) if 119903 = 1) and1199041+ 119904

2gt 0 then

1003817100381710038171003817119891119892

10038171003817100381710038171198611199041

119901119903

le 119862100381710038171003817100381711989110038171003817100381710038171198611199041

119901119903

100381710038171003817100381711989210038171003817100381710038171198611199042

119901119903

(20)

where 119862 are constants independent of 119891 119892

Then we present two related lemmas for the followingtransport equations

119891119905+ 119889 sdot nabla119891 = 119865

1198911003816100381610038161003816119905=0

= 1198910

(21)

where 119889 R times R119899rarr R119899 stands for a given time dependent

vector field and 1198910 R119899

rarr R119898 and 119865 R times R119899rarr R119898 are

known data

Lemma 12 (see [39 40]) Let 1 le 119901 le 1199011le infin 1 le 119903 le infin

and 1199011015840= 119901(119901 minus 1) Assume that 119904 gt minus119899 sdot min(1119901

1 1119901

1015840)

or 119904 gt minus1 minus 119899 sdot min(11199011 1119901

1015840) if nabla sdot 119889 = 0 Then there

exists a constant 119862 depending only on 119899 119901 1199011 119903 119904 such that

the following estimate holds true

10038171003817100381710038171198911003817100381710038171003817infin

119905([0119905]119861

119904

119901119903)

le 1198901198621int119905

0119885(120591)d120591

times [10038171003817100381710038171198910

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890minus1198621int120591

0119885(120585)119889120585

119865 (120591)119861119904

119901119903

d120591]

(22)

with

119885 (119905)

=

nabla119889 (119905)1198611198991199011

1199011infincap119871infin 119904 lt 1 +

119899

1199011

nabla119889 (119905)119861119904minus1

1199011119903

119904 gt 1+

119899

1199011

119900119903 119904=1+

119899

1199011

119903=1

(23)

If 119891 = 119889 then for all 119904 gt 0 (nabla sdot 119889 = 0 119904 gt minus1) (22) holds with119885(119905) = nabla119889(119905)

119871infin

Let us state the existence result for transport equationwith data in Besov space

Lemma 13 (see [40]) Let 119901 1199011 119903 119904 be as in the statement

of Lemma 12 and 1198910

isin 119861119904

119901119903and 119865 isin 119871

1([0 119879] 119861

119904

119901119903) 119889 isin

119871120588([0 119879] 119861

minus119872

infininfin) is a time dependent vector field for some

120588 gt 1119872 gt 0 such that if 119904 lt 1 + 1198991199011then nabla119889 isin

1198711([0 119879] 119861

1198991199011

1199011infin

cap 119871infin) if 119904 gt 1 + 119899119901

1or 119904 = 1 + 119899119901

1 119903 = 1

then nabla119889 isin 1198711([0 119879] 119861

119904minus1

1199011119903) Thus transport equation (21) has a

unique solution 119891 isin 119871infin([0 119879] 119861

119904

119901119903)⋂(⋂

1199041015840lt119904

119862([0 119879] 1198611199041015840

1199011))

and (22) holds true If 119903 lt infin then one has 119891 isin 119862([0 119879] 119861119904

119901119903)

3 The Proof of Theorem 1

We finish the proof of Theorem 1 by the following steps

Mathematical Problems in Engineering 5

31 Existence of Solutions For convenience we denote 120578 =

120588 minus 1 and rewrite (7) as

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205782

2

+ 120578]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

2119898119906119909119909

]

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909= minus119906

119909120578 minus 119906

119909 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(24)

We use a standard iterative process to construct the approxi-mate solutions to (24)

Step 1 Starting from 1199060

= 1205780

= 0 we define by inductiona sequence of smooth functions (119906

119894 120578

119894)119894isinN isin 119862(R+

119861infin

119901119903)2

solving the following transport equation

(120597119905+ 119906

119894120597119909) 119906

119894+1= 119865 (119905 119909) 119905 gt 0 119909 isin R

(120597119905+ 119906

119894120597119909) 120578

119894+1= minus120578

119894120597119909119906119894minus 120597

119909119906119894 119905 gt 0 119909 isin R

119906119894+1

(119909 0) = 119906119894+1

0(119909) = 119878

119894+11199060 119909 isin R

120578119894+1

(119909 0) = 120578119894+1

0(119909) = 119878

119894+11205780 119909 isin R

(25)

where

119865 (119905 119909) = 119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+ 120578119894]

+ 1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

(26)

Since all the data 119878119894+1

1199060 119878

119894+11205780isin 119861

infin

119901119903 Lemma 13 enables us

to show that for all 119894 isin N the system (25) has a global solutionwhich belongs to 119862(R+

119861infin

119901119903)2

Step 2 Next we prove that (119906119894 120578119894)119894isinN is uniformly bounded

in 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

According to Lemma 12 we have the following inequali-ties for all 119894 isin N10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

le 119890

1198621int119905

0119906119894(120591)119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890

minus1198621int120591

0119906119894(120585)119861119904

119901119903

d120585119865 (120591 sdot)

119861119904

119901119903

d120591]

(27)

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

1198622int119905

0119906119894119861119904

119901119903

d120591[10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198622int120591

0119906119894119861119904

119901119903

d120585

times

10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

d120591]

(28)

From Proposition 8 Lemma 11 and 119878minus1-multiplier property

of 119875(119863) it yields that1003817100381710038171003817100381710038171003817

119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+120578119894]

1003817100381710038171003817100381710038171003817119861119904

119901119903

le 119862(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

119861119904

119901119903

+2119896

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817

2

119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

(29)

Using the 119878minus2-multiplier property of 119875

1(119863) we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862

100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

(30)

(i) By the assumption of 119891(119906119894) = (119906

119894)2119899+1 and Lemma 9

we have100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119871infin)

(119904minus1)110038171003817100381710038171003817119891101584010038171003817100381710038171003817119882(119904minus1)1infin

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

(119904minus1)11003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(31)

(ii) As we know if max(32 1 + 1119901) lt 119904 le 2 + 1119901 then119861119904minus1

119901119903is an algebra if 119904 gt 2+1119901 then119861

119904minus2

119901119903is an algebra

By the assumption of ℎ(119906119894) = (119906119894)2119898 and Lemmas 10

and 11 we deduce that100381710038171003817100381710038171003817

120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862120573

100381710038171003817100381710038171003817

(119906119894)

2119898100381710038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

(119904minus1)1

times

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(32)

6 Mathematical Problems in Engineering

Combining (i) and (ii) yields100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(33)

Thanks to Lemma 11 we get10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(34)

Therefore from (27) to (34) we obtain10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 1198623sdot 119890

1198623int119905

0119906119894119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198623int120591

0119906119894119861119904

119901119903

d120585(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 119896 + 1)

1199041

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)] d120591

(35)

Let us choose a 119879 gt 0 such that 211990411198621199041+1

3( 119906

119894

0119861119904

119901119903

+ 120578119894

0119861119904minus1

119901119903

+

119896 + 1)1199041119879 lt 1 and

(119896 + 1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(36)

Inserting (36) into (35) yields

(119896 + 1 +

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(37)

Therefore (119906119894 120578119894)119894isinN is uniformly bounded in119862([0 119879] 119861

119904

119901119903)times

119862([0 119879] 119861119904minus1

119901119903) From Proposition 8 Lemma 11 and the fol-

lowing embedding properties

119861119904

119901119903997893rarr 119861

119904minus1

119901119903 119861

119904minus1

119901119903997893rarr 119861

119904minus2

119901119903 (38)

we have10038171003817100381710038171003817119906119894120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

(39)

We deduce that 119906119894120597119909119906119894+1 and 119865(119905 119909) are uniformly bounded

in 119862([0 119879] 119861119904minus1

119901119903) in the same way that 119906

119894120597119909120578119894+1

minus120578119894120597119909119906119894minus

120597119909119906119894 are uniformly bounded in 119862([0 119879] 119861

119904minus2

119901119903) Using the

system (25) we have (120597119905119906119894+1

120597119905120578119894+1

) isin 119862([0 119879] 119861119904minus1

119901119903) times

119862([0 119879] 119861119904minus2

119901119903) is uniformly bounded which derives that

(119906119894 120578

119894)119894isinN is uniformly bounded in 119864

119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

Step 3 Now we demonstrate that (119906119894 120578

119894)119894isinN is a Cauchy

sequence in 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

In fact according to (25) we note that for all 119894 119895 isin N wehave

(120597119905+ 119906

119894+119895120597119909) (119906

119894+119895+1minus 119906

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909119906119894+1

+ 119875 (119863)

times [(119906119894+119895

minus 119906119894) (119906

119894+119895+ 119906

119894) +

1

2

120597119909(119906

119894+119895minus 119906

119894)

sdot 120597119909(119906

119894+119895+ 119906

119894) + 2119896 (119906

119894+119895minus 119906

119894) +

1

2

(120578119894+119895

minus 120578119894)

times (120578119894+119895

+ 120578119894) + (120578

119894+119895minus 120578

119894) ]

+ 1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)

+ 120573 ((119906119894+119895

)

2119898

(119906119894+119895

)119909119909

minus (119906119894)

2119898

(119906119894)119909119909

)]

(40)

(120597119905+ 119906

119894+119895120597119909) (120578

119894+119895+1minus 120578

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909120578119894+1

minus (120578119894+119895

minus 120578119894) 120597

119909119906119894

minus 120578119894+119895

120597119909(119906

119894+119895minus 119906

119894) minus 120597

119909(119906

119894+119895minus 119906

119894)

(41)

(1) We estimate the terms in the right side of (40) ByLemma 11 we have10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (119906

119894+119895minus 119906

119894) (119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894+119895

+ 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [120597

119909(119906

119894+119895minus 119906

119894) 120597

119909(119906

119894+119895+ 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [2119896 (119906

119894+119895minus 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862119896

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817119875 (119863) [(120578

119894+119895minus 120578

119894) (120578

119894+119895+ 120578

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+119895

+ 12057811989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (120578

119894+119895minus 120578

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

(42)

Mathematical Problems in Engineering 7

We use Lemma 10 to estimate the high order terms in (40)Firstly we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

119904110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(43)

Secondly by 1199062119898

119906119909119909

= 120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909and

Lemma 11 we deduce that100381710038171003817100381710038171003817

120573 ((119906119894+119895

)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

le 119862(

100381710038171003817100381710038171003817

120597119909((119906

119894+119895)

2119898

119906119894+119895

119909minus (119906

119894)

2119898

119906119894

119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

+

100381710038171003817100381710038171003817

(119906119894+119895

)

2119898minus1

(119906119894+119895

119909)

2

minus (119906119894)

2119898minus1

(119906119894

119909)

2100381710038171003817100381710038171003817119861119904minus3

119901119903

)

(44)

Hence100381710038171003817100381710038171003817

1198751(119863) 120573 [(119906

119894+119895)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

1199041+110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(45)

(2) Now we estimate the right side of (41)10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817(120578

119894+119895minus 120578

119894) 120597

11990911990611989410038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

1003817100381710038171003817100381712059711990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120578119894+119895

120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(46)

Combining (1) (2) and Lemma 12 yields for all 119905 isin [0 119879]

10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817119906119894+119895+1

0minus 119906

119894+1

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862

times int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)(

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 1)

+

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+ 119896 + 1)

1199041+1

] d120591

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817120578119894+119895+1

0minus 120578

119894+1

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+ 1)

+(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

] d120591

(47)

Since (119906119894 120578119894)119894isinN is uniformly bounded in119864

119904

119901119903(119879)times119864

119904minus1

119901119903(119879) and

119906119894+119895+1

0minus 119906

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021199060

120578119894+119895+1

0minus 120578

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021205780

(48)

there exists a constant 119862119879independent of 119894 119895 such that for all

119905 isin [0 119879] we get10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862119879[2

minus119894+ int

119905

0

(

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

) d120591]

(49)

By induction we obtain that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le

(119862119879119879)

119894+1

(119894 + 1)

[

1003817100381710038171003817100381711990611989510038171003817100381710038171003817119871infin([0119879]119861

119904minus1

119901119903)+

1003817100381710038171003817100381712057811989510038171003817100381710038171003817119871infin([0119879]119861

119904minus2

119901119903)]

+ 119862119879

119894

sum

119897=0

2minus(119894minus119897)

(119862119879119879)

119897

119897

(50)

Since 119906119895119871infin

119879(119861119904

119901119903) 120578

119895119871infin

119879(119861119904minus1

119901119903)are bounded independently of

119895 we conclude that there exists a new constant 1198621198791

such that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le 1198621198791

2minus119894

(51)

Consequently (119906119894 120578119894)119894isinN is a Cauchy sequence in 119862([0 119879]

119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

8 Mathematical Problems in Engineering

Step 4 We end the proof of existence of solutions

Firstly since (119906119894 120578

119894)119894isinN is uniformly bounded in

119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) according to the Fatou

property for Besov space it guarantees that (119906 120578) alsobelongs to 119871

infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903)

Secondly for (119906119894 120578

119894)119894isinN is a Cauchy sequence in

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) so it converges to some

limit function (119906 120578) isin 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) An

interpolation argument insures that the convergence holdsin 119862([0 119879] 119861

1199041015840

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904 It is easy to

pass to the limit in (25) and to conclude that (119906 120578) is indeeda solution to (24) Thanks to the fact that (119906 120578) belongs to119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) we know that the right

side of the first equation in (24) belongs to 119871infin([0 119879] 119861

119904

119901119903)

and the right side of the second equation in (24) belongs to119871infin([0 119879] 119861

119904minus1

119901119903) In the case of 119903 lt infin applying Lemma 13

derives (119906 120578) isin 119862([0 119879] 1198611199041015840

119901119903) times119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904

Finally with (24) we obtain that (119906119905 120578

119905) isin

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) if 119903 lt infin and in

119871infin([0 119879] 119861

119904minus1

119901119903) times 119871

infin([0 119879] 119861

119904minus2

119901119903) otherwise Thus (119906 120578)

isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) Moreover a standard use of a sequence

of viscosity approximate solutions (119906120576 120578

120576)120576gt0

for (24) whichconverges uniformly in 119862([0 119879] 119861

119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) times

119862([0 119879] 119861119904minus1

119901119903) cap 119862

1([0 119879] 119861

119904minus2

119901119903) gives the continuity of

solution (119906 120578) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

32 Uniqueness and Continuity with Initial Data

Lemma 14 Assume that 1 le 119901 119903 le infin and 119904 gt max(1 +

1119901 32) Let (1199061 120578

1) and (119906

2 120578

2) be two given solutions to

the system (24) with initial data (1199061

0 120578

1

0) (119906

2

0 120578

2

0) isin 119861

119904

119901119903times

119861119904minus1

119901119903satisfying 119906

1 119906

2isin 119871

infin([0 119879] 119861

119904

119901119903) cap 119862([0 119879] 119861

119904minus1

119901119903) and

1205781 120578

2isin 119871

infin([0 119879] 119861

119904minus1

119901119903) cap 119862([0 119879] 119861

119904minus2

119901119903) Then for every

119905 isin [0 119879] one has

100381710038171003817100381710038171199061minus 119906

210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781minus 120578

210038171003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061

0minus 119906

2

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781

0minus 120578

2

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

times 119890

119862int119905

0[(1199061119861119904

119901119903

+1199062119861119904

119901119903

+119896+1)1199041+1

+1205781119861119904minus1

119901119903

+1205782119861119904minus1

119901119903

]d120591

(52)

Proof Denote 11990612

= 1199062minus 119906

1 120578

12= 120578

2minus 120578

1 then

11990612

isin 119871infin

([0 119879] 119861119904

119901119903) cap 119862 ([0 119879] 119861

119904minus1

119901119903)

12057812

isin 119871infin

([0 119879] 119861119904minus1

119901119903) cap 119862 ([0 119879] 119861

119904minus2

119901119903)

(53)

which implies that (11990612 120578

12) isin 119862([0 119879] 119861

119904minus1

119901119903) times 119862([0 119879]

119861119904minus2

119901119903) and (119906

12 120578

12) satisfies the following transport equation

12059711990511990612

+ 119906112059711990911990612

= minus119906121205971199091199062+ 119865

1(119905 119909)

119905 gt 0 119909 isin R

12059711990512057812

+ 119906112059711990912057812

= 1198650(119905 119909) 119905 gt 0 119909 isin R

11990612

(119909 0) = 11990612

0= 119906

2

0minus 119906

1

0 119909 isin R

12057812

(119909 0) = 12057812

0= 120578

2

0minus 120578

1

0 119909 isin R

(54)

with

1198651(119905 119909)

= 119875 (119863) [11990612

(1199061+ 119906

2) +

1

2

12059711990911990612120597119909(119906

1+ 119906

2)

+ 211989611990612

+

1

2

12057812

(1205781+ 120578

2+ 2)]

+ 1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

1198650(119905 119909) = minus119906

121205971199091205782minus (120578

121205971199091199061+ 120578

212059711990911990612

+ 12059711990911990612)

(55)

According to Lemma 12 we have

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585(

1003817100381710038171003817100381711990612120597119909119906210038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198651

1003817100381710038171003817B119904minus1119901119903

) d120591

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d12058510038171003817100381710038171198650

1003817100381710038171003817119861119904minus2

119901119903

d120591

(56)

Similar to the proof of the third step in Section 31 here weonly focus our attention on the high order terms We deducethat

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

(57)

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

(3) Complex interpolation

100381710038171003817100381711989110038171003817100381710038171198611205791199041+(1minus120579)1199042

119901119903

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

1198611199041

119901119903

10038171003817100381710038171198911003817100381710038171003817

1minus120579

1198611199042

119901119903

119891119900119903 119890V119890119903119910 119891 isin 1198611199041

119901119903cap 119861

1199042

119901119903 120579 isin [0 1]

(15)

(4) Fatoursquos Lemma if (119906119899)119899isinN is bounded in119861

119904

119901119903and 119906

119899rarr

119906 in 1198781015840(R) then 119906 isin 119861

119904

119901119903and

119906119861119904

119901119903

le lim119899rarrinfin

inf 1003817100381710038171003817119906119899

1003817100381710038171003817119861119904

119901119903

(16)

(5) Let 119898 isin R and 119891 be an 119878119898-multiplier (ie 119891 R119899

rarr

R is smooth and satisfies that for every 120572 isin N119899 thereexists a constant 119862

120572such that |120597

120572119891(120585)| le 119862

120572(1 +

|120585|)119898minus|120572| for all 120585 isin R119899) Then the operator 119891(119863) is

continuous from 119861119904

119901119903to 119861

119904minus119898

119901119903

(6) Density 119862infin

119888is dense in 119861

119904

119901119903hArr 1 le 119901 119903 lt infin

Lemma 9 (see [40]) Let 119868 be an open interval of R Let 119904 gt 0

and 1199041be the smallest integer such that 119904

1ge 119904 Let 119865 119868 rarr R

satisfy 119865(0) = 0 and 1198651015840isin 119882

1199041infin

(119868R) Assume that V isin 119861119904

119901119903

has values in 119869 subsub 119868 Then 119865(V) isin 119861119904

119901119903and there exists a

constant 119862 depending only on 119904 119868 119869119873 such that

119865 (V)119861119904

119901119903

le 119862(1 + V119871infin)

1199041100381710038171003817100381710038171198651015840100381710038171003817100381710038171198821199041infin(119868)

V119861119904

119901119903

(17)

Lemma 10 (see [40]) Let 119868 be an open interval of R Let119904 gt 0 and 119904

1be the smallest integer such that 119904

1ge 119904 Let

119865 119868 rarr R satisfy 1198651015840(0) = 0 and 119865

10158401015840isin 119882

1199041infin

(119868R) Assumethat 119906 V isin 119861

119904

119901119903has values in 119869 subsub 119868 Then there exists a

constant 119862 depending only on 119904 119868 119869119873 such that

119865 (119906) minus 119865 (V)119861119904

119901119903

le 119862(1 + 119906119871infin)

11990411003817100381710038171003817100381711986510158401015840100381710038171003817100381710038171198821199041infin(119868)

times (119906 minus V119861119904

119901119903

sup120579isin[01]

120579119906 + (1 minus 120579) V119871infin

+ 119906 minus V119871infin sup120579isin[01]

120579119906 + (1 minus 120579) V119861119904

119901119903

)

(18)

Lemma 11 (see [40]) Assume that 1 le 119901 and 119903 le infin thefollowing estimates hold

(i) for 119904 gt 0 then

1003817100381710038171003817119891119892

1003817100381710038171003817119861119904

119901119903

le 119862(10038171003817100381710038171198911003817100381710038171003817119861119904

119901119903

10038171003817100381710038171198921003817100381710038171003817119871infin +

10038171003817100381710038171198911003817100381710038171003817119871infin

10038171003817100381710038171198921003817100381710038171003817119861119904

119901119903

)

(19)

(ii) for 1199041

le 1119901 1199042

gt 1119901 (1199042

ge (1119901) if 119903 = 1) and1199041+ 119904

2gt 0 then

1003817100381710038171003817119891119892

10038171003817100381710038171198611199041

119901119903

le 119862100381710038171003817100381711989110038171003817100381710038171198611199041

119901119903

100381710038171003817100381711989210038171003817100381710038171198611199042

119901119903

(20)

where 119862 are constants independent of 119891 119892

Then we present two related lemmas for the followingtransport equations

119891119905+ 119889 sdot nabla119891 = 119865

1198911003816100381610038161003816119905=0

= 1198910

(21)

where 119889 R times R119899rarr R119899 stands for a given time dependent

vector field and 1198910 R119899

rarr R119898 and 119865 R times R119899rarr R119898 are

known data

Lemma 12 (see [39 40]) Let 1 le 119901 le 1199011le infin 1 le 119903 le infin

and 1199011015840= 119901(119901 minus 1) Assume that 119904 gt minus119899 sdot min(1119901

1 1119901

1015840)

or 119904 gt minus1 minus 119899 sdot min(11199011 1119901

1015840) if nabla sdot 119889 = 0 Then there

exists a constant 119862 depending only on 119899 119901 1199011 119903 119904 such that

the following estimate holds true

10038171003817100381710038171198911003817100381710038171003817infin

119905([0119905]119861

119904

119901119903)

le 1198901198621int119905

0119885(120591)d120591

times [10038171003817100381710038171198910

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890minus1198621int120591

0119885(120585)119889120585

119865 (120591)119861119904

119901119903

d120591]

(22)

with

119885 (119905)

=

nabla119889 (119905)1198611198991199011

1199011infincap119871infin 119904 lt 1 +

119899

1199011

nabla119889 (119905)119861119904minus1

1199011119903

119904 gt 1+

119899

1199011

119900119903 119904=1+

119899

1199011

119903=1

(23)

If 119891 = 119889 then for all 119904 gt 0 (nabla sdot 119889 = 0 119904 gt minus1) (22) holds with119885(119905) = nabla119889(119905)

119871infin

Let us state the existence result for transport equationwith data in Besov space

Lemma 13 (see [40]) Let 119901 1199011 119903 119904 be as in the statement

of Lemma 12 and 1198910

isin 119861119904

119901119903and 119865 isin 119871

1([0 119879] 119861

119904

119901119903) 119889 isin

119871120588([0 119879] 119861

minus119872

infininfin) is a time dependent vector field for some

120588 gt 1119872 gt 0 such that if 119904 lt 1 + 1198991199011then nabla119889 isin

1198711([0 119879] 119861

1198991199011

1199011infin

cap 119871infin) if 119904 gt 1 + 119899119901

1or 119904 = 1 + 119899119901

1 119903 = 1

then nabla119889 isin 1198711([0 119879] 119861

119904minus1

1199011119903) Thus transport equation (21) has a

unique solution 119891 isin 119871infin([0 119879] 119861

119904

119901119903)⋂(⋂

1199041015840lt119904

119862([0 119879] 1198611199041015840

1199011))

and (22) holds true If 119903 lt infin then one has 119891 isin 119862([0 119879] 119861119904

119901119903)

3 The Proof of Theorem 1

We finish the proof of Theorem 1 by the following steps

Mathematical Problems in Engineering 5

31 Existence of Solutions For convenience we denote 120578 =

120588 minus 1 and rewrite (7) as

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205782

2

+ 120578]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

2119898119906119909119909

]

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909= minus119906

119909120578 minus 119906

119909 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(24)

We use a standard iterative process to construct the approxi-mate solutions to (24)

Step 1 Starting from 1199060

= 1205780

= 0 we define by inductiona sequence of smooth functions (119906

119894 120578

119894)119894isinN isin 119862(R+

119861infin

119901119903)2

solving the following transport equation

(120597119905+ 119906

119894120597119909) 119906

119894+1= 119865 (119905 119909) 119905 gt 0 119909 isin R

(120597119905+ 119906

119894120597119909) 120578

119894+1= minus120578

119894120597119909119906119894minus 120597

119909119906119894 119905 gt 0 119909 isin R

119906119894+1

(119909 0) = 119906119894+1

0(119909) = 119878

119894+11199060 119909 isin R

120578119894+1

(119909 0) = 120578119894+1

0(119909) = 119878

119894+11205780 119909 isin R

(25)

where

119865 (119905 119909) = 119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+ 120578119894]

+ 1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

(26)

Since all the data 119878119894+1

1199060 119878

119894+11205780isin 119861

infin

119901119903 Lemma 13 enables us

to show that for all 119894 isin N the system (25) has a global solutionwhich belongs to 119862(R+

119861infin

119901119903)2

Step 2 Next we prove that (119906119894 120578119894)119894isinN is uniformly bounded

in 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

According to Lemma 12 we have the following inequali-ties for all 119894 isin N10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

le 119890

1198621int119905

0119906119894(120591)119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890

minus1198621int120591

0119906119894(120585)119861119904

119901119903

d120585119865 (120591 sdot)

119861119904

119901119903

d120591]

(27)

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

1198622int119905

0119906119894119861119904

119901119903

d120591[10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198622int120591

0119906119894119861119904

119901119903

d120585

times

10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

d120591]

(28)

From Proposition 8 Lemma 11 and 119878minus1-multiplier property

of 119875(119863) it yields that1003817100381710038171003817100381710038171003817

119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+120578119894]

1003817100381710038171003817100381710038171003817119861119904

119901119903

le 119862(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

119861119904

119901119903

+2119896

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817

2

119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

(29)

Using the 119878minus2-multiplier property of 119875

1(119863) we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862

100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

(30)

(i) By the assumption of 119891(119906119894) = (119906

119894)2119899+1 and Lemma 9

we have100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119871infin)

(119904minus1)110038171003817100381710038171003817119891101584010038171003817100381710038171003817119882(119904minus1)1infin

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

(119904minus1)11003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(31)

(ii) As we know if max(32 1 + 1119901) lt 119904 le 2 + 1119901 then119861119904minus1

119901119903is an algebra if 119904 gt 2+1119901 then119861

119904minus2

119901119903is an algebra

By the assumption of ℎ(119906119894) = (119906119894)2119898 and Lemmas 10

and 11 we deduce that100381710038171003817100381710038171003817

120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862120573

100381710038171003817100381710038171003817

(119906119894)

2119898100381710038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

(119904minus1)1

times

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(32)

6 Mathematical Problems in Engineering

Combining (i) and (ii) yields100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(33)

Thanks to Lemma 11 we get10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(34)

Therefore from (27) to (34) we obtain10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 1198623sdot 119890

1198623int119905

0119906119894119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198623int120591

0119906119894119861119904

119901119903

d120585(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 119896 + 1)

1199041

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)] d120591

(35)

Let us choose a 119879 gt 0 such that 211990411198621199041+1

3( 119906

119894

0119861119904

119901119903

+ 120578119894

0119861119904minus1

119901119903

+

119896 + 1)1199041119879 lt 1 and

(119896 + 1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(36)

Inserting (36) into (35) yields

(119896 + 1 +

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(37)

Therefore (119906119894 120578119894)119894isinN is uniformly bounded in119862([0 119879] 119861

119904

119901119903)times

119862([0 119879] 119861119904minus1

119901119903) From Proposition 8 Lemma 11 and the fol-

lowing embedding properties

119861119904

119901119903997893rarr 119861

119904minus1

119901119903 119861

119904minus1

119901119903997893rarr 119861

119904minus2

119901119903 (38)

we have10038171003817100381710038171003817119906119894120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

(39)

We deduce that 119906119894120597119909119906119894+1 and 119865(119905 119909) are uniformly bounded

in 119862([0 119879] 119861119904minus1

119901119903) in the same way that 119906

119894120597119909120578119894+1

minus120578119894120597119909119906119894minus

120597119909119906119894 are uniformly bounded in 119862([0 119879] 119861

119904minus2

119901119903) Using the

system (25) we have (120597119905119906119894+1

120597119905120578119894+1

) isin 119862([0 119879] 119861119904minus1

119901119903) times

119862([0 119879] 119861119904minus2

119901119903) is uniformly bounded which derives that

(119906119894 120578

119894)119894isinN is uniformly bounded in 119864

119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

Step 3 Now we demonstrate that (119906119894 120578

119894)119894isinN is a Cauchy

sequence in 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

In fact according to (25) we note that for all 119894 119895 isin N wehave

(120597119905+ 119906

119894+119895120597119909) (119906

119894+119895+1minus 119906

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909119906119894+1

+ 119875 (119863)

times [(119906119894+119895

minus 119906119894) (119906

119894+119895+ 119906

119894) +

1

2

120597119909(119906

119894+119895minus 119906

119894)

sdot 120597119909(119906

119894+119895+ 119906

119894) + 2119896 (119906

119894+119895minus 119906

119894) +

1

2

(120578119894+119895

minus 120578119894)

times (120578119894+119895

+ 120578119894) + (120578

119894+119895minus 120578

119894) ]

+ 1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)

+ 120573 ((119906119894+119895

)

2119898

(119906119894+119895

)119909119909

minus (119906119894)

2119898

(119906119894)119909119909

)]

(40)

(120597119905+ 119906

119894+119895120597119909) (120578

119894+119895+1minus 120578

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909120578119894+1

minus (120578119894+119895

minus 120578119894) 120597

119909119906119894

minus 120578119894+119895

120597119909(119906

119894+119895minus 119906

119894) minus 120597

119909(119906

119894+119895minus 119906

119894)

(41)

(1) We estimate the terms in the right side of (40) ByLemma 11 we have10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (119906

119894+119895minus 119906

119894) (119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894+119895

+ 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [120597

119909(119906

119894+119895minus 119906

119894) 120597

119909(119906

119894+119895+ 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [2119896 (119906

119894+119895minus 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862119896

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817119875 (119863) [(120578

119894+119895minus 120578

119894) (120578

119894+119895+ 120578

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+119895

+ 12057811989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (120578

119894+119895minus 120578

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

(42)

Mathematical Problems in Engineering 7

We use Lemma 10 to estimate the high order terms in (40)Firstly we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

119904110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(43)

Secondly by 1199062119898

119906119909119909

= 120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909and

Lemma 11 we deduce that100381710038171003817100381710038171003817

120573 ((119906119894+119895

)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

le 119862(

100381710038171003817100381710038171003817

120597119909((119906

119894+119895)

2119898

119906119894+119895

119909minus (119906

119894)

2119898

119906119894

119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

+

100381710038171003817100381710038171003817

(119906119894+119895

)

2119898minus1

(119906119894+119895

119909)

2

minus (119906119894)

2119898minus1

(119906119894

119909)

2100381710038171003817100381710038171003817119861119904minus3

119901119903

)

(44)

Hence100381710038171003817100381710038171003817

1198751(119863) 120573 [(119906

119894+119895)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

1199041+110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(45)

(2) Now we estimate the right side of (41)10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817(120578

119894+119895minus 120578

119894) 120597

11990911990611989410038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

1003817100381710038171003817100381712059711990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120578119894+119895

120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(46)

Combining (1) (2) and Lemma 12 yields for all 119905 isin [0 119879]

10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817119906119894+119895+1

0minus 119906

119894+1

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862

times int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)(

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 1)

+

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+ 119896 + 1)

1199041+1

] d120591

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817120578119894+119895+1

0minus 120578

119894+1

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+ 1)

+(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

] d120591

(47)

Since (119906119894 120578119894)119894isinN is uniformly bounded in119864

119904

119901119903(119879)times119864

119904minus1

119901119903(119879) and

119906119894+119895+1

0minus 119906

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021199060

120578119894+119895+1

0minus 120578

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021205780

(48)

there exists a constant 119862119879independent of 119894 119895 such that for all

119905 isin [0 119879] we get10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862119879[2

minus119894+ int

119905

0

(

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

) d120591]

(49)

By induction we obtain that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le

(119862119879119879)

119894+1

(119894 + 1)

[

1003817100381710038171003817100381711990611989510038171003817100381710038171003817119871infin([0119879]119861

119904minus1

119901119903)+

1003817100381710038171003817100381712057811989510038171003817100381710038171003817119871infin([0119879]119861

119904minus2

119901119903)]

+ 119862119879

119894

sum

119897=0

2minus(119894minus119897)

(119862119879119879)

119897

119897

(50)

Since 119906119895119871infin

119879(119861119904

119901119903) 120578

119895119871infin

119879(119861119904minus1

119901119903)are bounded independently of

119895 we conclude that there exists a new constant 1198621198791

such that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le 1198621198791

2minus119894

(51)

Consequently (119906119894 120578119894)119894isinN is a Cauchy sequence in 119862([0 119879]

119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

8 Mathematical Problems in Engineering

Step 4 We end the proof of existence of solutions

Firstly since (119906119894 120578

119894)119894isinN is uniformly bounded in

119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) according to the Fatou

property for Besov space it guarantees that (119906 120578) alsobelongs to 119871

infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903)

Secondly for (119906119894 120578

119894)119894isinN is a Cauchy sequence in

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) so it converges to some

limit function (119906 120578) isin 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) An

interpolation argument insures that the convergence holdsin 119862([0 119879] 119861

1199041015840

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904 It is easy to

pass to the limit in (25) and to conclude that (119906 120578) is indeeda solution to (24) Thanks to the fact that (119906 120578) belongs to119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) we know that the right

side of the first equation in (24) belongs to 119871infin([0 119879] 119861

119904

119901119903)

and the right side of the second equation in (24) belongs to119871infin([0 119879] 119861

119904minus1

119901119903) In the case of 119903 lt infin applying Lemma 13

derives (119906 120578) isin 119862([0 119879] 1198611199041015840

119901119903) times119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904

Finally with (24) we obtain that (119906119905 120578

119905) isin

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) if 119903 lt infin and in

119871infin([0 119879] 119861

119904minus1

119901119903) times 119871

infin([0 119879] 119861

119904minus2

119901119903) otherwise Thus (119906 120578)

isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) Moreover a standard use of a sequence

of viscosity approximate solutions (119906120576 120578

120576)120576gt0

for (24) whichconverges uniformly in 119862([0 119879] 119861

119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) times

119862([0 119879] 119861119904minus1

119901119903) cap 119862

1([0 119879] 119861

119904minus2

119901119903) gives the continuity of

solution (119906 120578) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

32 Uniqueness and Continuity with Initial Data

Lemma 14 Assume that 1 le 119901 119903 le infin and 119904 gt max(1 +

1119901 32) Let (1199061 120578

1) and (119906

2 120578

2) be two given solutions to

the system (24) with initial data (1199061

0 120578

1

0) (119906

2

0 120578

2

0) isin 119861

119904

119901119903times

119861119904minus1

119901119903satisfying 119906

1 119906

2isin 119871

infin([0 119879] 119861

119904

119901119903) cap 119862([0 119879] 119861

119904minus1

119901119903) and

1205781 120578

2isin 119871

infin([0 119879] 119861

119904minus1

119901119903) cap 119862([0 119879] 119861

119904minus2

119901119903) Then for every

119905 isin [0 119879] one has

100381710038171003817100381710038171199061minus 119906

210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781minus 120578

210038171003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061

0minus 119906

2

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781

0minus 120578

2

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

times 119890

119862int119905

0[(1199061119861119904

119901119903

+1199062119861119904

119901119903

+119896+1)1199041+1

+1205781119861119904minus1

119901119903

+1205782119861119904minus1

119901119903

]d120591

(52)

Proof Denote 11990612

= 1199062minus 119906

1 120578

12= 120578

2minus 120578

1 then

11990612

isin 119871infin

([0 119879] 119861119904

119901119903) cap 119862 ([0 119879] 119861

119904minus1

119901119903)

12057812

isin 119871infin

([0 119879] 119861119904minus1

119901119903) cap 119862 ([0 119879] 119861

119904minus2

119901119903)

(53)

which implies that (11990612 120578

12) isin 119862([0 119879] 119861

119904minus1

119901119903) times 119862([0 119879]

119861119904minus2

119901119903) and (119906

12 120578

12) satisfies the following transport equation

12059711990511990612

+ 119906112059711990911990612

= minus119906121205971199091199062+ 119865

1(119905 119909)

119905 gt 0 119909 isin R

12059711990512057812

+ 119906112059711990912057812

= 1198650(119905 119909) 119905 gt 0 119909 isin R

11990612

(119909 0) = 11990612

0= 119906

2

0minus 119906

1

0 119909 isin R

12057812

(119909 0) = 12057812

0= 120578

2

0minus 120578

1

0 119909 isin R

(54)

with

1198651(119905 119909)

= 119875 (119863) [11990612

(1199061+ 119906

2) +

1

2

12059711990911990612120597119909(119906

1+ 119906

2)

+ 211989611990612

+

1

2

12057812

(1205781+ 120578

2+ 2)]

+ 1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

1198650(119905 119909) = minus119906

121205971199091205782minus (120578

121205971199091199061+ 120578

212059711990911990612

+ 12059711990911990612)

(55)

According to Lemma 12 we have

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585(

1003817100381710038171003817100381711990612120597119909119906210038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198651

1003817100381710038171003817B119904minus1119901119903

) d120591

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d12058510038171003817100381710038171198650

1003817100381710038171003817119861119904minus2

119901119903

d120591

(56)

Similar to the proof of the third step in Section 31 here weonly focus our attention on the high order terms We deducethat

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

(57)

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

31 Existence of Solutions For convenience we denote 120578 =

120588 minus 1 and rewrite (7) as

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205782

2

+ 120578]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

2119898119906119909119909

]

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909= minus119906

119909120578 minus 119906

119909 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(24)

We use a standard iterative process to construct the approxi-mate solutions to (24)

Step 1 Starting from 1199060

= 1205780

= 0 we define by inductiona sequence of smooth functions (119906

119894 120578

119894)119894isinN isin 119862(R+

119861infin

119901119903)2

solving the following transport equation

(120597119905+ 119906

119894120597119909) 119906

119894+1= 119865 (119905 119909) 119905 gt 0 119909 isin R

(120597119905+ 119906

119894120597119909) 120578

119894+1= minus120578

119894120597119909119906119894minus 120597

119909119906119894 119905 gt 0 119909 isin R

119906119894+1

(119909 0) = 119906119894+1

0(119909) = 119878

119894+11199060 119909 isin R

120578119894+1

(119909 0) = 120578119894+1

0(119909) = 119878

119894+11205780 119909 isin R

(25)

where

119865 (119905 119909) = 119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+ 120578119894]

+ 1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

(26)

Since all the data 119878119894+1

1199060 119878

119894+11205780isin 119861

infin

119901119903 Lemma 13 enables us

to show that for all 119894 isin N the system (25) has a global solutionwhich belongs to 119862(R+

119861infin

119901119903)2

Step 2 Next we prove that (119906119894 120578119894)119894isinN is uniformly bounded

in 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

According to Lemma 12 we have the following inequali-ties for all 119894 isin N10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

le 119890

1198621int119905

0119906119894(120591)119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+ int

119905

0

119890

minus1198621int120591

0119906119894(120585)119861119904

119901119903

d120585119865 (120591 sdot)

119861119904

119901119903

d120591]

(27)

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

1198622int119905

0119906119894119861119904

119901119903

d120591[10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198622int120591

0119906119894119861119904

119901119903

d120585

times

10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

d120591]

(28)

From Proposition 8 Lemma 11 and 119878minus1-multiplier property

of 119875(119863) it yields that1003817100381710038171003817100381710038171003817

119875 (119863) [(119906119894)

2

+

1

2

(119906119894)

2

119909+ 2119896119906

119894+

1

2

(120578119894)

2

+120578119894]

1003817100381710038171003817100381710038171003817119861119904

119901119903

le 119862(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817

2

119861119904

119901119903

+2119896

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817

2

119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

(29)

Using the 119878minus2-multiplier property of 119875

1(119863) we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862

100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1

+ 120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

(30)

(i) By the assumption of 119891(119906119894) = (119906

119894)2119899+1 and Lemma 9

we have100381710038171003817100381710038171003817

minus120582(119906119894)

2119899+1100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119871infin)

(119904minus1)110038171003817100381710038171003817119891101584010038171003817100381710038171003817119882(119904minus1)1infin

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

(119904minus1)11003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(31)

(ii) As we know if max(32 1 + 1119901) lt 119904 le 2 + 1119901 then119861119904minus1

119901119903is an algebra if 119904 gt 2+1119901 then119861

119904minus2

119901119903is an algebra

By the assumption of ℎ(119906119894) = (119906119894)2119898 and Lemmas 10

and 11 we deduce that100381710038171003817100381710038171003817

120573(119906119894)

2119898

119906119894

119909119909

100381710038171003817100381710038171003817119861119904minus2

119901119903

le 119862120573

100381710038171003817100381710038171003817

(119906119894)

2119898100381710038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

(119904minus1)1

times

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894

119909119909

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(32)

6 Mathematical Problems in Engineering

Combining (i) and (ii) yields100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(33)

Thanks to Lemma 11 we get10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(34)

Therefore from (27) to (34) we obtain10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 1198623sdot 119890

1198623int119905

0119906119894119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198623int120591

0119906119894119861119904

119901119903

d120585(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 119896 + 1)

1199041

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)] d120591

(35)

Let us choose a 119879 gt 0 such that 211990411198621199041+1

3( 119906

119894

0119861119904

119901119903

+ 120578119894

0119861119904minus1

119901119903

+

119896 + 1)1199041119879 lt 1 and

(119896 + 1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(36)

Inserting (36) into (35) yields

(119896 + 1 +

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(37)

Therefore (119906119894 120578119894)119894isinN is uniformly bounded in119862([0 119879] 119861

119904

119901119903)times

119862([0 119879] 119861119904minus1

119901119903) From Proposition 8 Lemma 11 and the fol-

lowing embedding properties

119861119904

119901119903997893rarr 119861

119904minus1

119901119903 119861

119904minus1

119901119903997893rarr 119861

119904minus2

119901119903 (38)

we have10038171003817100381710038171003817119906119894120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

(39)

We deduce that 119906119894120597119909119906119894+1 and 119865(119905 119909) are uniformly bounded

in 119862([0 119879] 119861119904minus1

119901119903) in the same way that 119906

119894120597119909120578119894+1

minus120578119894120597119909119906119894minus

120597119909119906119894 are uniformly bounded in 119862([0 119879] 119861

119904minus2

119901119903) Using the

system (25) we have (120597119905119906119894+1

120597119905120578119894+1

) isin 119862([0 119879] 119861119904minus1

119901119903) times

119862([0 119879] 119861119904minus2

119901119903) is uniformly bounded which derives that

(119906119894 120578

119894)119894isinN is uniformly bounded in 119864

119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

Step 3 Now we demonstrate that (119906119894 120578

119894)119894isinN is a Cauchy

sequence in 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

In fact according to (25) we note that for all 119894 119895 isin N wehave

(120597119905+ 119906

119894+119895120597119909) (119906

119894+119895+1minus 119906

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909119906119894+1

+ 119875 (119863)

times [(119906119894+119895

minus 119906119894) (119906

119894+119895+ 119906

119894) +

1

2

120597119909(119906

119894+119895minus 119906

119894)

sdot 120597119909(119906

119894+119895+ 119906

119894) + 2119896 (119906

119894+119895minus 119906

119894) +

1

2

(120578119894+119895

minus 120578119894)

times (120578119894+119895

+ 120578119894) + (120578

119894+119895minus 120578

119894) ]

+ 1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)

+ 120573 ((119906119894+119895

)

2119898

(119906119894+119895

)119909119909

minus (119906119894)

2119898

(119906119894)119909119909

)]

(40)

(120597119905+ 119906

119894+119895120597119909) (120578

119894+119895+1minus 120578

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909120578119894+1

minus (120578119894+119895

minus 120578119894) 120597

119909119906119894

minus 120578119894+119895

120597119909(119906

119894+119895minus 119906

119894) minus 120597

119909(119906

119894+119895minus 119906

119894)

(41)

(1) We estimate the terms in the right side of (40) ByLemma 11 we have10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (119906

119894+119895minus 119906

119894) (119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894+119895

+ 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [120597

119909(119906

119894+119895minus 119906

119894) 120597

119909(119906

119894+119895+ 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [2119896 (119906

119894+119895minus 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862119896

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817119875 (119863) [(120578

119894+119895minus 120578

119894) (120578

119894+119895+ 120578

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+119895

+ 12057811989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (120578

119894+119895minus 120578

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

(42)

Mathematical Problems in Engineering 7

We use Lemma 10 to estimate the high order terms in (40)Firstly we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

119904110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(43)

Secondly by 1199062119898

119906119909119909

= 120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909and

Lemma 11 we deduce that100381710038171003817100381710038171003817

120573 ((119906119894+119895

)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

le 119862(

100381710038171003817100381710038171003817

120597119909((119906

119894+119895)

2119898

119906119894+119895

119909minus (119906

119894)

2119898

119906119894

119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

+

100381710038171003817100381710038171003817

(119906119894+119895

)

2119898minus1

(119906119894+119895

119909)

2

minus (119906119894)

2119898minus1

(119906119894

119909)

2100381710038171003817100381710038171003817119861119904minus3

119901119903

)

(44)

Hence100381710038171003817100381710038171003817

1198751(119863) 120573 [(119906

119894+119895)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

1199041+110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(45)

(2) Now we estimate the right side of (41)10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817(120578

119894+119895minus 120578

119894) 120597

11990911990611989410038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

1003817100381710038171003817100381712059711990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120578119894+119895

120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(46)

Combining (1) (2) and Lemma 12 yields for all 119905 isin [0 119879]

10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817119906119894+119895+1

0minus 119906

119894+1

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862

times int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)(

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 1)

+

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+ 119896 + 1)

1199041+1

] d120591

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817120578119894+119895+1

0minus 120578

119894+1

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+ 1)

+(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

] d120591

(47)

Since (119906119894 120578119894)119894isinN is uniformly bounded in119864

119904

119901119903(119879)times119864

119904minus1

119901119903(119879) and

119906119894+119895+1

0minus 119906

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021199060

120578119894+119895+1

0minus 120578

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021205780

(48)

there exists a constant 119862119879independent of 119894 119895 such that for all

119905 isin [0 119879] we get10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862119879[2

minus119894+ int

119905

0

(

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

) d120591]

(49)

By induction we obtain that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le

(119862119879119879)

119894+1

(119894 + 1)

[

1003817100381710038171003817100381711990611989510038171003817100381710038171003817119871infin([0119879]119861

119904minus1

119901119903)+

1003817100381710038171003817100381712057811989510038171003817100381710038171003817119871infin([0119879]119861

119904minus2

119901119903)]

+ 119862119879

119894

sum

119897=0

2minus(119894minus119897)

(119862119879119879)

119897

119897

(50)

Since 119906119895119871infin

119879(119861119904

119901119903) 120578

119895119871infin

119879(119861119904minus1

119901119903)are bounded independently of

119895 we conclude that there exists a new constant 1198621198791

such that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le 1198621198791

2minus119894

(51)

Consequently (119906119894 120578119894)119894isinN is a Cauchy sequence in 119862([0 119879]

119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

8 Mathematical Problems in Engineering

Step 4 We end the proof of existence of solutions

Firstly since (119906119894 120578

119894)119894isinN is uniformly bounded in

119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) according to the Fatou

property for Besov space it guarantees that (119906 120578) alsobelongs to 119871

infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903)

Secondly for (119906119894 120578

119894)119894isinN is a Cauchy sequence in

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) so it converges to some

limit function (119906 120578) isin 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) An

interpolation argument insures that the convergence holdsin 119862([0 119879] 119861

1199041015840

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904 It is easy to

pass to the limit in (25) and to conclude that (119906 120578) is indeeda solution to (24) Thanks to the fact that (119906 120578) belongs to119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) we know that the right

side of the first equation in (24) belongs to 119871infin([0 119879] 119861

119904

119901119903)

and the right side of the second equation in (24) belongs to119871infin([0 119879] 119861

119904minus1

119901119903) In the case of 119903 lt infin applying Lemma 13

derives (119906 120578) isin 119862([0 119879] 1198611199041015840

119901119903) times119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904

Finally with (24) we obtain that (119906119905 120578

119905) isin

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) if 119903 lt infin and in

119871infin([0 119879] 119861

119904minus1

119901119903) times 119871

infin([0 119879] 119861

119904minus2

119901119903) otherwise Thus (119906 120578)

isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) Moreover a standard use of a sequence

of viscosity approximate solutions (119906120576 120578

120576)120576gt0

for (24) whichconverges uniformly in 119862([0 119879] 119861

119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) times

119862([0 119879] 119861119904minus1

119901119903) cap 119862

1([0 119879] 119861

119904minus2

119901119903) gives the continuity of

solution (119906 120578) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

32 Uniqueness and Continuity with Initial Data

Lemma 14 Assume that 1 le 119901 119903 le infin and 119904 gt max(1 +

1119901 32) Let (1199061 120578

1) and (119906

2 120578

2) be two given solutions to

the system (24) with initial data (1199061

0 120578

1

0) (119906

2

0 120578

2

0) isin 119861

119904

119901119903times

119861119904minus1

119901119903satisfying 119906

1 119906

2isin 119871

infin([0 119879] 119861

119904

119901119903) cap 119862([0 119879] 119861

119904minus1

119901119903) and

1205781 120578

2isin 119871

infin([0 119879] 119861

119904minus1

119901119903) cap 119862([0 119879] 119861

119904minus2

119901119903) Then for every

119905 isin [0 119879] one has

100381710038171003817100381710038171199061minus 119906

210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781minus 120578

210038171003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061

0minus 119906

2

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781

0minus 120578

2

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

times 119890

119862int119905

0[(1199061119861119904

119901119903

+1199062119861119904

119901119903

+119896+1)1199041+1

+1205781119861119904minus1

119901119903

+1205782119861119904minus1

119901119903

]d120591

(52)

Proof Denote 11990612

= 1199062minus 119906

1 120578

12= 120578

2minus 120578

1 then

11990612

isin 119871infin

([0 119879] 119861119904

119901119903) cap 119862 ([0 119879] 119861

119904minus1

119901119903)

12057812

isin 119871infin

([0 119879] 119861119904minus1

119901119903) cap 119862 ([0 119879] 119861

119904minus2

119901119903)

(53)

which implies that (11990612 120578

12) isin 119862([0 119879] 119861

119904minus1

119901119903) times 119862([0 119879]

119861119904minus2

119901119903) and (119906

12 120578

12) satisfies the following transport equation

12059711990511990612

+ 119906112059711990911990612

= minus119906121205971199091199062+ 119865

1(119905 119909)

119905 gt 0 119909 isin R

12059711990512057812

+ 119906112059711990912057812

= 1198650(119905 119909) 119905 gt 0 119909 isin R

11990612

(119909 0) = 11990612

0= 119906

2

0minus 119906

1

0 119909 isin R

12057812

(119909 0) = 12057812

0= 120578

2

0minus 120578

1

0 119909 isin R

(54)

with

1198651(119905 119909)

= 119875 (119863) [11990612

(1199061+ 119906

2) +

1

2

12059711990911990612120597119909(119906

1+ 119906

2)

+ 211989611990612

+

1

2

12057812

(1205781+ 120578

2+ 2)]

+ 1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

1198650(119905 119909) = minus119906

121205971199091205782minus (120578

121205971199091199061+ 120578

212059711990911990612

+ 12059711990911990612)

(55)

According to Lemma 12 we have

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585(

1003817100381710038171003817100381711990612120597119909119906210038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198651

1003817100381710038171003817B119904minus1119901119903

) d120591

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d12058510038171003817100381710038171198650

1003817100381710038171003817119861119904minus2

119901119903

d120591

(56)

Similar to the proof of the third step in Section 31 here weonly focus our attention on the high order terms We deducethat

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

(57)

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Combining (i) and (ii) yields100381710038171003817100381710038171003817

1198751(119863) [minus120582(119906

119894)

2119899+1

+ 120573(119906119894)

2119898

(119906119894)119909119909

]

100381710038171003817100381710038171003817119861119904

119901119903

le 119862(1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

11990411003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(33)

Thanks to Lemma 11 we get10038171003817100381710038171003817minus120578

119894120597119909119906119894minus 120597

11990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

(34)

Therefore from (27) to (34) we obtain10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 1198623sdot 119890

1198623int119905

0119906119894119861119904

119901119903

d120591

times [10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

+ int

119905

0

119890

minus1198623int120591

0119906119894119861119904

119901119903

d120585(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 119896 + 1)

1199041

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)] d120591

(35)

Let us choose a 119879 gt 0 such that 211990411198621199041+1

3( 119906

119894

0119861119904

119901119903

+ 120578119894

0119861119904minus1

119901119903

+

119896 + 1)1199041119879 lt 1 and

(119896 + 1 +

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(36)

Inserting (36) into (35) yields

(119896 + 1 +

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

)

1199041

le

1198621199041

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

1 minus 211990411198621199041+1

3(119896 + 1 +

10038171003817100381710038171199060

1003817100381710038171003817119861119904

119901119903

+10038171003817100381710038171205780

1003817100381710038171003817119861119904minus1

119901119903

)

1199041

119905

(37)

Therefore (119906119894 120578119894)119894isinN is uniformly bounded in119862([0 119879] 119861

119904

119901119903)times

119862([0 119879] 119861119904minus1

119901119903) From Proposition 8 Lemma 11 and the fol-

lowing embedding properties

119861119904

119901119903997893rarr 119861

119904minus1

119901119903 119861

119904minus1

119901119903997893rarr 119861

119904minus2

119901119903 (38)

we have10038171003817100381710038171003817119906119894120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

10038171003817100381710038171003817119906119894120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

(39)

We deduce that 119906119894120597119909119906119894+1 and 119865(119905 119909) are uniformly bounded

in 119862([0 119879] 119861119904minus1

119901119903) in the same way that 119906

119894120597119909120578119894+1

minus120578119894120597119909119906119894minus

120597119909119906119894 are uniformly bounded in 119862([0 119879] 119861

119904minus2

119901119903) Using the

system (25) we have (120597119905119906119894+1

120597119905120578119894+1

) isin 119862([0 119879] 119861119904minus1

119901119903) times

119862([0 119879] 119861119904minus2

119901119903) is uniformly bounded which derives that

(119906119894 120578

119894)119894isinN is uniformly bounded in 119864

119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

Step 3 Now we demonstrate that (119906119894 120578

119894)119894isinN is a Cauchy

sequence in 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

In fact according to (25) we note that for all 119894 119895 isin N wehave

(120597119905+ 119906

119894+119895120597119909) (119906

119894+119895+1minus 119906

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909119906119894+1

+ 119875 (119863)

times [(119906119894+119895

minus 119906119894) (119906

119894+119895+ 119906

119894) +

1

2

120597119909(119906

119894+119895minus 119906

119894)

sdot 120597119909(119906

119894+119895+ 119906

119894) + 2119896 (119906

119894+119895minus 119906

119894) +

1

2

(120578119894+119895

minus 120578119894)

times (120578119894+119895

+ 120578119894) + (120578

119894+119895minus 120578

119894) ]

+ 1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)

+ 120573 ((119906119894+119895

)

2119898

(119906119894+119895

)119909119909

minus (119906119894)

2119898

(119906119894)119909119909

)]

(40)

(120597119905+ 119906

119894+119895120597119909) (120578

119894+119895+1minus 120578

119894+1)

= (119906119894minus 119906

119894+119895) 120597

119909120578119894+1

minus (120578119894+119895

minus 120578119894) 120597

119909119906119894

minus 120578119894+119895

120597119909(119906

119894+119895minus 119906

119894) minus 120597

119909(119906

119894+119895minus 119906

119894)

(41)

(1) We estimate the terms in the right side of (40) ByLemma 11 we have10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (119906

119894+119895minus 119906

119894) (119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119906119894+119895

+ 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [120597

119909(119906

119894+119895minus 119906

119894) 120597

119909(119906

119894+119895+ 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895+ 119906

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) [2119896 (119906

119894+119895minus 119906

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862119896

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817119875 (119863) [(120578

119894+119895minus 120578

119894) (120578

119894+119895+ 120578

119894)]

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+119895

+ 12057811989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817119875 (119863) (120578

119894+119895minus 120578

119894)

10038171003817100381710038171003817119861119904minus1

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

(42)

Mathematical Problems in Engineering 7

We use Lemma 10 to estimate the high order terms in (40)Firstly we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

119904110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(43)

Secondly by 1199062119898

119906119909119909

= 120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909and

Lemma 11 we deduce that100381710038171003817100381710038171003817

120573 ((119906119894+119895

)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

le 119862(

100381710038171003817100381710038171003817

120597119909((119906

119894+119895)

2119898

119906119894+119895

119909minus (119906

119894)

2119898

119906119894

119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

+

100381710038171003817100381710038171003817

(119906119894+119895

)

2119898minus1

(119906119894+119895

119909)

2

minus (119906119894)

2119898minus1

(119906119894

119909)

2100381710038171003817100381710038171003817119861119904minus3

119901119903

)

(44)

Hence100381710038171003817100381710038171003817

1198751(119863) 120573 [(119906

119894+119895)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

1199041+110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(45)

(2) Now we estimate the right side of (41)10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817(120578

119894+119895minus 120578

119894) 120597

11990911990611989410038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

1003817100381710038171003817100381712059711990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120578119894+119895

120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(46)

Combining (1) (2) and Lemma 12 yields for all 119905 isin [0 119879]

10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817119906119894+119895+1

0minus 119906

119894+1

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862

times int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)(

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 1)

+

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+ 119896 + 1)

1199041+1

] d120591

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817120578119894+119895+1

0minus 120578

119894+1

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+ 1)

+(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

] d120591

(47)

Since (119906119894 120578119894)119894isinN is uniformly bounded in119864

119904

119901119903(119879)times119864

119904minus1

119901119903(119879) and

119906119894+119895+1

0minus 119906

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021199060

120578119894+119895+1

0minus 120578

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021205780

(48)

there exists a constant 119862119879independent of 119894 119895 such that for all

119905 isin [0 119879] we get10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862119879[2

minus119894+ int

119905

0

(

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

) d120591]

(49)

By induction we obtain that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le

(119862119879119879)

119894+1

(119894 + 1)

[

1003817100381710038171003817100381711990611989510038171003817100381710038171003817119871infin([0119879]119861

119904minus1

119901119903)+

1003817100381710038171003817100381712057811989510038171003817100381710038171003817119871infin([0119879]119861

119904minus2

119901119903)]

+ 119862119879

119894

sum

119897=0

2minus(119894minus119897)

(119862119879119879)

119897

119897

(50)

Since 119906119895119871infin

119879(119861119904

119901119903) 120578

119895119871infin

119879(119861119904minus1

119901119903)are bounded independently of

119895 we conclude that there exists a new constant 1198621198791

such that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le 1198621198791

2minus119894

(51)

Consequently (119906119894 120578119894)119894isinN is a Cauchy sequence in 119862([0 119879]

119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

8 Mathematical Problems in Engineering

Step 4 We end the proof of existence of solutions

Firstly since (119906119894 120578

119894)119894isinN is uniformly bounded in

119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) according to the Fatou

property for Besov space it guarantees that (119906 120578) alsobelongs to 119871

infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903)

Secondly for (119906119894 120578

119894)119894isinN is a Cauchy sequence in

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) so it converges to some

limit function (119906 120578) isin 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) An

interpolation argument insures that the convergence holdsin 119862([0 119879] 119861

1199041015840

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904 It is easy to

pass to the limit in (25) and to conclude that (119906 120578) is indeeda solution to (24) Thanks to the fact that (119906 120578) belongs to119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) we know that the right

side of the first equation in (24) belongs to 119871infin([0 119879] 119861

119904

119901119903)

and the right side of the second equation in (24) belongs to119871infin([0 119879] 119861

119904minus1

119901119903) In the case of 119903 lt infin applying Lemma 13

derives (119906 120578) isin 119862([0 119879] 1198611199041015840

119901119903) times119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904

Finally with (24) we obtain that (119906119905 120578

119905) isin

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) if 119903 lt infin and in

119871infin([0 119879] 119861

119904minus1

119901119903) times 119871

infin([0 119879] 119861

119904minus2

119901119903) otherwise Thus (119906 120578)

isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) Moreover a standard use of a sequence

of viscosity approximate solutions (119906120576 120578

120576)120576gt0

for (24) whichconverges uniformly in 119862([0 119879] 119861

119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) times

119862([0 119879] 119861119904minus1

119901119903) cap 119862

1([0 119879] 119861

119904minus2

119901119903) gives the continuity of

solution (119906 120578) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

32 Uniqueness and Continuity with Initial Data

Lemma 14 Assume that 1 le 119901 119903 le infin and 119904 gt max(1 +

1119901 32) Let (1199061 120578

1) and (119906

2 120578

2) be two given solutions to

the system (24) with initial data (1199061

0 120578

1

0) (119906

2

0 120578

2

0) isin 119861

119904

119901119903times

119861119904minus1

119901119903satisfying 119906

1 119906

2isin 119871

infin([0 119879] 119861

119904

119901119903) cap 119862([0 119879] 119861

119904minus1

119901119903) and

1205781 120578

2isin 119871

infin([0 119879] 119861

119904minus1

119901119903) cap 119862([0 119879] 119861

119904minus2

119901119903) Then for every

119905 isin [0 119879] one has

100381710038171003817100381710038171199061minus 119906

210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781minus 120578

210038171003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061

0minus 119906

2

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781

0minus 120578

2

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

times 119890

119862int119905

0[(1199061119861119904

119901119903

+1199062119861119904

119901119903

+119896+1)1199041+1

+1205781119861119904minus1

119901119903

+1205782119861119904minus1

119901119903

]d120591

(52)

Proof Denote 11990612

= 1199062minus 119906

1 120578

12= 120578

2minus 120578

1 then

11990612

isin 119871infin

([0 119879] 119861119904

119901119903) cap 119862 ([0 119879] 119861

119904minus1

119901119903)

12057812

isin 119871infin

([0 119879] 119861119904minus1

119901119903) cap 119862 ([0 119879] 119861

119904minus2

119901119903)

(53)

which implies that (11990612 120578

12) isin 119862([0 119879] 119861

119904minus1

119901119903) times 119862([0 119879]

119861119904minus2

119901119903) and (119906

12 120578

12) satisfies the following transport equation

12059711990511990612

+ 119906112059711990911990612

= minus119906121205971199091199062+ 119865

1(119905 119909)

119905 gt 0 119909 isin R

12059711990512057812

+ 119906112059711990912057812

= 1198650(119905 119909) 119905 gt 0 119909 isin R

11990612

(119909 0) = 11990612

0= 119906

2

0minus 119906

1

0 119909 isin R

12057812

(119909 0) = 12057812

0= 120578

2

0minus 120578

1

0 119909 isin R

(54)

with

1198651(119905 119909)

= 119875 (119863) [11990612

(1199061+ 119906

2) +

1

2

12059711990911990612120597119909(119906

1+ 119906

2)

+ 211989611990612

+

1

2

12057812

(1205781+ 120578

2+ 2)]

+ 1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

1198650(119905 119909) = minus119906

121205971199091205782minus (120578

121205971199091199061+ 120578

212059711990911990612

+ 12059711990911990612)

(55)

According to Lemma 12 we have

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585(

1003817100381710038171003817100381711990612120597119909119906210038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198651

1003817100381710038171003817B119904minus1119901119903

) d120591

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d12058510038171003817100381710038171198650

1003817100381710038171003817119861119904minus2

119901119903

d120591

(56)

Similar to the proof of the third step in Section 31 here weonly focus our attention on the high order terms We deducethat

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

(57)

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

We use Lemma 10 to estimate the high order terms in (40)Firstly we get

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

119894+119895)

2119899+1

minus (119906119894)

2119899+1

)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904minus1

119901119903

)

119904110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(43)

Secondly by 1199062119898

119906119909119909

= 120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909and

Lemma 11 we deduce that100381710038171003817100381710038171003817

120573 ((119906119894+119895

)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

le 119862(

100381710038171003817100381710038171003817

120597119909((119906

119894+119895)

2119898

119906119894+119895

119909minus (119906

119894)

2119898

119906119894

119909)

100381710038171003817100381710038171003817119861119904minus3

119901119903

+

100381710038171003817100381710038171003817

(119906119894+119895

)

2119898minus1

(119906119894+119895

119909)

2

minus (119906119894)

2119898minus1

(119906119894

119909)

2100381710038171003817100381710038171003817119861119904minus3

119901119903

)

(44)

Hence100381710038171003817100381710038171003817

1198751(119863) 120573 [(119906

119894+119895)

2119898

119906119894+119895

119909119909minus (119906

119894)

2119898

119906119894

119909119909]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

)

1199041+110038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(45)

(2) Now we estimate the right side of (41)10038171003817100381710038171003817(119906

119894minus 119906

119894+119895) 120597

119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817(120578

119894+119895minus 120578

119894) 120597

11990911990611989410038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

1003817100381710038171003817100381712059711990911990611989410038171003817100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120578119894+119895

120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

10038171003817100381710038171003817120597119909(119906

119894+119895minus 119906

119894)

10038171003817100381710038171003817119861119904minus2

119901119903

le 119862

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(46)

Combining (1) (2) and Lemma 12 yields for all 119905 isin [0 119879]

10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817119906119894+119895+1

0minus 119906

119894+1

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862

times int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)(

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057811989410038171003817100381710038171003817119861119904minus1

119901119903

+ 1)

+

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

times(

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+11989510038171003817

100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906119894+110038171003817

100381710038171003817119861119904

119901119903

+ 119896 + 1)

1199041+1

] d120591

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119890

119862int119905

0119906119894+119895119861119904

119901119903

d120591

times

10038171003817100381710038171003817120578119894+119895+1

0minus 120578

119894+1

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

0119906119894+119895119861119904

119901119903

d120585

times [

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

(

10038171003817100381710038171003817120578119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+11989510038171003817

100381710038171003817119861119904minus1

119901119903

+ 1)

+(

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

)

1003817100381710038171003817100381711990611989410038171003817100381710038171003817119861119904

119901119903

] d120591

(47)

Since (119906119894 120578119894)119894isinN is uniformly bounded in119864

119904

119901119903(119879)times119864

119904minus1

119901119903(119879) and

119906119894+119895+1

0minus 119906

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021199060

120578119894+119895+1

0minus 120578

119894+1

0=

119894+119895

sum

119902=119894+1

Δ1199021205780

(48)

there exists a constant 119862119879independent of 119894 119895 such that for all

119905 isin [0 119879] we get10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119861119904minus2

119901119903

le 119862119879[2

minus119894+ int

119905

0

(

10038171003817100381710038171003817119906119894+119895

minus 11990611989410038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578119894+119895

minus 12057811989410038171003817100381710038171003817119861119904minus2

119901119903

) d120591]

(49)

By induction we obtain that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le

(119862119879119879)

119894+1

(119894 + 1)

[

1003817100381710038171003817100381711990611989510038171003817100381710038171003817119871infin([0119879]119861

119904minus1

119901119903)+

1003817100381710038171003817100381712057811989510038171003817100381710038171003817119871infin([0119879]119861

119904minus2

119901119903)]

+ 119862119879

119894

sum

119897=0

2minus(119894minus119897)

(119862119879119879)

119897

119897

(50)

Since 119906119895119871infin

119879(119861119904

119901119903) 120578

119895119871infin

119879(119861119904minus1

119901119903)are bounded independently of

119895 we conclude that there exists a new constant 1198621198791

such that10038171003817100381710038171003817119906119894+119895+1

minus 119906119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus1

119901119903)+

10038171003817100381710038171003817120578119894+119895+1

minus 120578119894+110038171003817

100381710038171003817119871infin

119879(119861119904minus2

119901119903)

le 1198621198791

2minus119894

(51)

Consequently (119906119894 120578119894)119894isinN is a Cauchy sequence in 119862([0 119879]

119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903)

8 Mathematical Problems in Engineering

Step 4 We end the proof of existence of solutions

Firstly since (119906119894 120578

119894)119894isinN is uniformly bounded in

119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) according to the Fatou

property for Besov space it guarantees that (119906 120578) alsobelongs to 119871

infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903)

Secondly for (119906119894 120578

119894)119894isinN is a Cauchy sequence in

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) so it converges to some

limit function (119906 120578) isin 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) An

interpolation argument insures that the convergence holdsin 119862([0 119879] 119861

1199041015840

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904 It is easy to

pass to the limit in (25) and to conclude that (119906 120578) is indeeda solution to (24) Thanks to the fact that (119906 120578) belongs to119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) we know that the right

side of the first equation in (24) belongs to 119871infin([0 119879] 119861

119904

119901119903)

and the right side of the second equation in (24) belongs to119871infin([0 119879] 119861

119904minus1

119901119903) In the case of 119903 lt infin applying Lemma 13

derives (119906 120578) isin 119862([0 119879] 1198611199041015840

119901119903) times119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904

Finally with (24) we obtain that (119906119905 120578

119905) isin

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) if 119903 lt infin and in

119871infin([0 119879] 119861

119904minus1

119901119903) times 119871

infin([0 119879] 119861

119904minus2

119901119903) otherwise Thus (119906 120578)

isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) Moreover a standard use of a sequence

of viscosity approximate solutions (119906120576 120578

120576)120576gt0

for (24) whichconverges uniformly in 119862([0 119879] 119861

119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) times

119862([0 119879] 119861119904minus1

119901119903) cap 119862

1([0 119879] 119861

119904minus2

119901119903) gives the continuity of

solution (119906 120578) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

32 Uniqueness and Continuity with Initial Data

Lemma 14 Assume that 1 le 119901 119903 le infin and 119904 gt max(1 +

1119901 32) Let (1199061 120578

1) and (119906

2 120578

2) be two given solutions to

the system (24) with initial data (1199061

0 120578

1

0) (119906

2

0 120578

2

0) isin 119861

119904

119901119903times

119861119904minus1

119901119903satisfying 119906

1 119906

2isin 119871

infin([0 119879] 119861

119904

119901119903) cap 119862([0 119879] 119861

119904minus1

119901119903) and

1205781 120578

2isin 119871

infin([0 119879] 119861

119904minus1

119901119903) cap 119862([0 119879] 119861

119904minus2

119901119903) Then for every

119905 isin [0 119879] one has

100381710038171003817100381710038171199061minus 119906

210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781minus 120578

210038171003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061

0minus 119906

2

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781

0minus 120578

2

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

times 119890

119862int119905

0[(1199061119861119904

119901119903

+1199062119861119904

119901119903

+119896+1)1199041+1

+1205781119861119904minus1

119901119903

+1205782119861119904minus1

119901119903

]d120591

(52)

Proof Denote 11990612

= 1199062minus 119906

1 120578

12= 120578

2minus 120578

1 then

11990612

isin 119871infin

([0 119879] 119861119904

119901119903) cap 119862 ([0 119879] 119861

119904minus1

119901119903)

12057812

isin 119871infin

([0 119879] 119861119904minus1

119901119903) cap 119862 ([0 119879] 119861

119904minus2

119901119903)

(53)

which implies that (11990612 120578

12) isin 119862([0 119879] 119861

119904minus1

119901119903) times 119862([0 119879]

119861119904minus2

119901119903) and (119906

12 120578

12) satisfies the following transport equation

12059711990511990612

+ 119906112059711990911990612

= minus119906121205971199091199062+ 119865

1(119905 119909)

119905 gt 0 119909 isin R

12059711990512057812

+ 119906112059711990912057812

= 1198650(119905 119909) 119905 gt 0 119909 isin R

11990612

(119909 0) = 11990612

0= 119906

2

0minus 119906

1

0 119909 isin R

12057812

(119909 0) = 12057812

0= 120578

2

0minus 120578

1

0 119909 isin R

(54)

with

1198651(119905 119909)

= 119875 (119863) [11990612

(1199061+ 119906

2) +

1

2

12059711990911990612120597119909(119906

1+ 119906

2)

+ 211989611990612

+

1

2

12057812

(1205781+ 120578

2+ 2)]

+ 1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

1198650(119905 119909) = minus119906

121205971199091205782minus (120578

121205971199091199061+ 120578

212059711990911990612

+ 12059711990911990612)

(55)

According to Lemma 12 we have

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585(

1003817100381710038171003817100381711990612120597119909119906210038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198651

1003817100381710038171003817B119904minus1119901119903

) d120591

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d12058510038171003817100381710038171198650

1003817100381710038171003817119861119904minus2

119901119903

d120591

(56)

Similar to the proof of the third step in Section 31 here weonly focus our attention on the high order terms We deducethat

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

(57)

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Step 4 We end the proof of existence of solutions

Firstly since (119906119894 120578

119894)119894isinN is uniformly bounded in

119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) according to the Fatou

property for Besov space it guarantees that (119906 120578) alsobelongs to 119871

infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903)

Secondly for (119906119894 120578

119894)119894isinN is a Cauchy sequence in

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) so it converges to some

limit function (119906 120578) isin 119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) An

interpolation argument insures that the convergence holdsin 119862([0 119879] 119861

1199041015840

119901119903) times 119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904 It is easy to

pass to the limit in (25) and to conclude that (119906 120578) is indeeda solution to (24) Thanks to the fact that (119906 120578) belongs to119871infin([0 119879] 119861

119904

119901119903) times 119871

infin([0 119879] 119861

119904minus1

119901119903) we know that the right

side of the first equation in (24) belongs to 119871infin([0 119879] 119861

119904

119901119903)

and the right side of the second equation in (24) belongs to119871infin([0 119879] 119861

119904minus1

119901119903) In the case of 119903 lt infin applying Lemma 13

derives (119906 120578) isin 119862([0 119879] 1198611199041015840

119901119903) times119862([0 119879] 119861

1199041015840minus1

119901119903) for any 119904

1015840lt 119904

Finally with (24) we obtain that (119906119905 120578

119905) isin

119862([0 119879] 119861119904minus1

119901119903) times 119862([0 119879] 119861

119904minus2

119901119903) if 119903 lt infin and in

119871infin([0 119879] 119861

119904minus1

119901119903) times 119871

infin([0 119879] 119861

119904minus2

119901119903) otherwise Thus (119906 120578)

isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879) Moreover a standard use of a sequence

of viscosity approximate solutions (119906120576 120578

120576)120576gt0

for (24) whichconverges uniformly in 119862([0 119879] 119861

119904

119901119903) cap 119862

1([0 119879] 119861

119904minus1

119901119903) times

119862([0 119879] 119861119904minus1

119901119903) cap 119862

1([0 119879] 119861

119904minus2

119901119903) gives the continuity of

solution (119906 120578) isin 119864119904

119901119903(119879) times 119864

119904minus1

119901119903(119879)

32 Uniqueness and Continuity with Initial Data

Lemma 14 Assume that 1 le 119901 119903 le infin and 119904 gt max(1 +

1119901 32) Let (1199061 120578

1) and (119906

2 120578

2) be two given solutions to

the system (24) with initial data (1199061

0 120578

1

0) (119906

2

0 120578

2

0) isin 119861

119904

119901119903times

119861119904minus1

119901119903satisfying 119906

1 119906

2isin 119871

infin([0 119879] 119861

119904

119901119903) cap 119862([0 119879] 119861

119904minus1

119901119903) and

1205781 120578

2isin 119871

infin([0 119879] 119861

119904minus1

119901119903) cap 119862([0 119879] 119861

119904minus2

119901119903) Then for every

119905 isin [0 119879] one has

100381710038171003817100381710038171199061minus 119906

210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781minus 120578

210038171003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061

0minus 119906

2

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781

0minus 120578

2

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

times 119890

119862int119905

0[(1199061119861119904

119901119903

+1199062119861119904

119901119903

+119896+1)1199041+1

+1205781119861119904minus1

119901119903

+1205782119861119904minus1

119901119903

]d120591

(52)

Proof Denote 11990612

= 1199062minus 119906

1 120578

12= 120578

2minus 120578

1 then

11990612

isin 119871infin

([0 119879] 119861119904

119901119903) cap 119862 ([0 119879] 119861

119904minus1

119901119903)

12057812

isin 119871infin

([0 119879] 119861119904minus1

119901119903) cap 119862 ([0 119879] 119861

119904minus2

119901119903)

(53)

which implies that (11990612 120578

12) isin 119862([0 119879] 119861

119904minus1

119901119903) times 119862([0 119879]

119861119904minus2

119901119903) and (119906

12 120578

12) satisfies the following transport equation

12059711990511990612

+ 119906112059711990911990612

= minus119906121205971199091199062+ 119865

1(119905 119909)

119905 gt 0 119909 isin R

12059711990512057812

+ 119906112059711990912057812

= 1198650(119905 119909) 119905 gt 0 119909 isin R

11990612

(119909 0) = 11990612

0= 119906

2

0minus 119906

1

0 119909 isin R

12057812

(119909 0) = 12057812

0= 120578

2

0minus 120578

1

0 119909 isin R

(54)

with

1198651(119905 119909)

= 119875 (119863) [11990612

(1199061+ 119906

2) +

1

2

12059711990911990612120597119909(119906

1+ 119906

2)

+ 211989611990612

+

1

2

12057812

(1205781+ 120578

2+ 2)]

+ 1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

1198650(119905 119909) = minus119906

121205971199091205782minus (120578

121205971199091199061+ 120578

212059711990911990612

+ 12059711990911990612)

(55)

According to Lemma 12 we have

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

le

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585(

1003817100381710038171003817100381711990612120597119909119906210038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198651

1003817100381710038171003817B119904minus1119901119903

) d120591

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

le

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d12058510038171003817100381710038171198650

1003817100381710038171003817119861119904minus2

119901119903

d120591

(56)

Similar to the proof of the third step in Section 31 here weonly focus our attention on the high order terms We deducethat

100381710038171003817100381710038171003817

1198751(119863) [minus120582 ((119906

1)

2119899+1

minus (1199062)

2119899+1

)

+ 120573 ((1199061)

2119898

1199061

119909119909minus (119906

2)

2119898

1199062

119909119909)]

100381710038171003817100381710038171003817119861119904minus1

119901119903

le 119862(1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

(57)

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Hence10038171003817100381710038171003817minus119906

121205971199091199062+ 119865

1(119905 119909)

10038171003817100381710038171003817119861119904minus1

119901119903

+10038171003817100381710038171198650(119905 119909)

1003817100381710038171003817119861119904minus2

119901119903

le (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

]

(58)

which together with (56) makes us obtain

119890

minus119862int119905

01205971199091199061119861119904minus1

119901119903

d120591(

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

le (

1003817100381710038171003817100381711990612

0

10038171003817100381710038171003817119861119904minus1

119901119903

+

1003817100381710038171003817100381712057812

0

10038171003817100381710038171003817119861119904minus2

119901119903

)

+ 119862int

119905

0

119890

minus119862int120591

01205971199091199061119861119904minus1

119901119903

d120585sdot (

100381710038171003817100381710038171199061210038171003817100381710038171003817119861119904minus1

119901119903

+

100381710038171003817100381710038171205781210038171003817100381710038171003817119861119904minus2

119901119903

)

times [(119896 + 1 +

10038171003817100381710038171003817119906110038171003817100381710038171003817119861119904

119901119903

+

10038171003817100381710038171003817119906210038171003817100381710038171003817119861119904

119901119903

)

1199041+1

+

10038171003817100381710038171003817120578110038171003817100381710038171003817119861119904minus1

119901119903

+

10038171003817100381710038171003817120578210038171003817100381710038171003817119861119904minus1

119901119903

] d120591

(59)

Using Gronwallrsquos inequality we complete the proof ofLemma 14

Remark 15 When 119901 = 119903 = 2 the Besov space 119861119904

119901119903(R)

coincides with the Sobolev space 119867119904(R) Theorem 1 implies

that under the assumption (1199060 120588

0minus 1) isin 119867

s(R) times 119867

119904minus1(R)

with 119904 gt 32 we obtain the local well-posedness for system(1) which improves the corresponding results in [20 25 35]by Katorsquos theory where 119904 ge 2 is assumed in [25 35] and 119904 ge 3

in [20]

Remark 16 The existence time for system (1) may be chosenindependently of 119904 in the following sense [43] If (119906 120588 minus

1) isin 119862([0 119879]119867119904) cap 119862

1([0 119879]119867

119904minus1) times 119862([0 119879]119867

119904minus1) cap

1198621([0 119879]119867

119904minus2) is the solution of system (1) with initial data

(1199060 120588

0minus 1) isin 119867

119903times H119903minus1 for some 119903 gt 32 119903 = 119904 then

(119906 120588minus1) isin 119862([0 119879]119867119903)cap119862

1([0 119879]119867

119903minus1)times119862([0 119879]119867

119903minus1)cap

1198621([0 119879]119867

119903minus2) with the same time 119879 In particular if (119906 120588 minus

1) isin 119867infin

times 119867infin then (119906 120588 minus 1) isin 119862([0 119879]119867

infin) times

119862([0 119879]119867infin)

4 Wave-Breaking Phenomena

In this section attention is turned to investigating conditionsof wave breaking and the precise blow-up rate of strongsolutions to system (1) Using Theorem 1 and Remark 16we establish the precise wave-breaking scenarios of strongsolutions to system (1)

41 The Proof ofTheorem 2 ApplyingTheorem 1 Remark 16and a simple density argument we only need to show thatTheorem 2 holds with 119904 ge 3 Here we assume 119904 = 3 to provethe theorem

Noting that 120578 = 120588 minus 1 we rewrite the system (1) as

119906119905minus 119906

119909119909119905+ 3119906119906

119909+ 120578120578

119909+ 120578

119909+ 2119896119906

119909

+ 1205821199062119899+1

minus 1205731199062119898

119906119909119909

= 2119906119909119906119909119909

+ 119906119906119909119909119909

119905 gt 0 119909 isin R

120578119905+ 119906120578

119909+ 120578119906

119909+ 119906

119909= 0 119905 gt 0 119909 isin R

119906 (119909 0) = 1199060(119909) 119909 isin R

120578 (119909 0) = 1205780(119909) = 120588

0(119909) minus 1 119909 isin R

(60)

Multiplying the first equation in (60) by 119906 and integrating byparts we deduce that

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 + int

R

1205821199062119899+2d119909

+ int

R

120573 (2119898 + 1) 1199062119898

1199062

119909d119909

=

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(61)

Since intR120582119906

2119899+2d119909 ge 0 and intR120573(2119898 + 1)119906

21198981199062

119909d119909 ge 0 then

1

2

dd119905

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

int

R

1205782119906119909d119909 + int

R

120578119906119909d119909

(62)

Multiplying the second equation in (60) by 120578 and integratingby parts we have

1

2

dd119905

int

R

1205782d119909 = minus

1

2

int

R

1205782119906119909d119909 minus int

R

120578119906119909d119909

(63)

which together with (62) impliesdd119905

int

R

(1199062+ 119906

2

119909+ 120578

2) d119909 le 0 (64)

On the other hand multiplying the first equation in (60) by119906119909119909

and integrating by parts again we get

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573 int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909) d119909

+

1

2

int

R

119906119909[(120578

2)119909119909

+ 2120578119909119909

] d119909

(65)

Multiplying the second equation in (60) by 120578119909119909

and integrat-ing by parts we obtain

minus

1

2

dd119905

int

R

1205782

119909d119909 =

1

2

int

R

119906119909(120578

2

119909minus 2120578120578

119909119909minus 2120578

119909119909) d119909 (66)

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

which together with (65) derives

minus

1

2

dd119905

int

R

(1199062

119909+ 119906

2

119909119909) d119909 minus

1

2

dd119905

int

R

1205782

119909d119909 minus 120582 (2119899 + 1)

times int

R

(11990621198991199062

119909) d119909 minus 120573int

R

(1199062119898

1199062

119909119909) d119909

=

3

2

int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(67)

For 120582(2119899 + 1) intR(119906

21198991199062

119909)d119909 ge 0 and 120573int

R(119906

21198981199062

119909119909)d119909 ge 0 we

have

dd119905

int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909 le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(68)

Therefore combining (64) with (68) one deduces that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le minus3int

R

119906119909(119906

2

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(69)

Assume that 119879 le +infin and there exists119872 gt 0 such that

119906119909(119905 119909) ge minus119872 for (119905 119909) isin [0 119879) timesR

1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin le 119872 for 119905 isin [0 119879)

(70)

It follows from (69) that

dd119905

int

R

(1199062+ 2119906

2

119909+ 119906

2

119909119909+ 120578

2+ 120578

2

119909) d119909

le 3119872int

R

(1199062

119909+ 119906

2

119909119909+ 120578

2

119909) d119909

(71)

Applying Gronwallrsquos inequality to (71) yields for all 119905 isin [0 119879)

119906 (119905)2

1198672 +

1003817100381710038171003817120578 (119905)

1003817100381710038171003817

2

1198671 le 2 (

10038171003817100381710038171199060

1003817100381710038171003817

2

1198672 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198671) 119890

3119872119879 (72)

Differentiating the first equation in (60) with respect to 119909 andmultiplying the obtained equation by 119906

119909119909119909 then integrating

by parts we get1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909) d119909

=

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times int

R

1199062119899minus2

1199064

119909d119909 minus 120582 (2119899 + 1) int

R

11990621198991199062

119909119909d119909

+ 120573119898 (2119898 minus 1)int

R

1199062119898minus2

1199062

1199091199091199062

119909d119909

+ 120573119898int

R

1199063

1199091199091199062119898minus1d119909 minus 120573int

R

1199062119898

1199062

119909119909119909d119909 minus

15

2

times int

R

1199061199091199062

119909119909d119909 minus

5

2

int

R

1199061199091199062

119909119909119909d119909

minus 2int

R

120578119909120578119909119909

119906119909119909d119909 + int

R

120578120578119909119909

119906119909119909119909

d119909

+ int

R

120578119909119909

119906119909119909119909

d119909

(73)

Differentiating the second equation twice in (60) with respectto 119909 and multiplying the resulted equation by 120578

119909119909 then

integrating by parts we deduce that1

2

dd119905

int

R

1205782

119909119909d119909 = minus

5

2

int

R

1199061199091205782

119909119909d119909 minus 3int

R

120578119909120578119909119909

119906119909119909d119909

minus int

R

120578120578119909119909

119906119909119909119909

d119909 minus int

R

120578119909119909

119906119909119909119909

d119909(74)

which together with (73) 120582(2119899 + 1) intR11990621198991199062

119909119909d119909 ge 0 and

120573intR1199062119898

1199062

119909119909119909d119909 ge 0 one has

1

2

dd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 10119872int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

+

120582 (2119899 minus 1) 2119899 (2119899 + 1)

3

times1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119899minus2

119871infin

times int

R

1199062

119909d119909 + 120573119898 (2119898 minus 1)

1003817100381710038171003817119906119909

1003817100381710038171003817

2

119871infin119906

2119898minus2

119871infin

times int

R

1199062

119909119909d119909 + 120573119898119906

2119898minus1

119871infin int

R

1199063

119909119909d119909

(75)

where we have used (64) (70) and (72)Applying Sobolevrsquos embedding theorem and Holderrsquos

inequality yields

int

R

1199063

119909119909d119909 le

1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119871infin int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909

le1003817100381710038171003817119906119909119909

1003817100381710038171003817

2

119867(12)+ int

R

1003816100381610038161003816119906119909119909

1003816100381610038161003816d119909 le 119906

2

1198673119906119867

2

(76)

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

together with (72) and (75) thus there exists1198621gt 0 and119862

2gt

0 such thatdd119905

int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

le 1198621+ 119862

2int

R

(1199062

119909119909+ 119906

2

119909119909119909+ 120578

2

119909119909) d119909

(77)

Using Gronwallrsquos inequality for all 119905 isin (0 119879) we have

int

R

(1199062

119909119909+ 119906

2

119909119909x + 1205782

119909119909) d119909

le 1198901198622119905int

R

(1199062

0119909119909+ 119906

2

0119909119909119909+ 120578

2

0119909119909) d119909 +

1198621

1198622

(1198901198622119905minus 1)

(78)

which contradicts the assumption of the maximal existencetime 119879 lt +infin

Conversely using Sobolevrsquos embedding theorem 119867119904997893rarr

119871infin

(119904 gt 12) we derive that if condition (9) in Theorem 2holds the corresponding solution blows up in finite timewhich completes the proof of Theorem 2

42 The Proof of Theorem 3 Firstly we present three lemmaswhich are used to prove the blow-up mechanisms and theglobal existence of solutions to system (1)

Lemma 17 (see [44]) Let 119879 gt 0 and 119906 isin 1198621([0 119879)119867

2(R))

Then for all 119905 isin [0 119879) there exists at least one point 120585(119905) isin R

with119898(119905) = inf

119909isinR119906119909(119905 119909) = 119906

119909(119905 120585 (119905)) (79)

The function 119898(119905) is absolutely continuous on (0 119879) with

ddt

119898(119905) = 119906119909119905

(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (80)

Now consider the following trajectory equation

ddt

119902 (119905 119909) = 119906 (119905 119902 (119905 119909)) 119905 isin [0 119879)

119902 (119909 0) = 119909 119909 isin R

(81)

where 119906 denotes the first component of solution (119906 120588 minus 1) tosystem (1)

Lemma 18 (see [45]) Let 119906 isin 119862([0 119879)119867119904(R)) cap

1198621([0 119879)119867

119904minus1(R)) with 119904 ge 2 Then (81) has a unique

solution 119902 isin 1198621([0 119879) times RR) Moreover the map 119902(119905 sdot) is

an increasing diffeomorphism of R for every 119905 isin [0 119879) and

119902119909(119905 119909) = 119890

int119905

0119906119909(120591119902(120591119909))d120591

gt 0 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(82)

Lemma 19 (see [23]) Let (1199060 120588

0minus 1) isin 119867

119904times 119867

119904minus1 with119904 ge 2 and let 119879 gt 0 be the maximal existence time of thecorresponding solution (119906 120588 minus 1) to system (1) Thus

120588 (119905 119902 (119905 119909)) 119902119909(119905 119909) = 120588

0(119909) 119891119900119903 119886119897119897 (119905 119909) isin [0 119879) timesR

(83)

Proof of Theorem 3 The technique used here is inspired from[20] Similar to the proof ofTheorem 2 we only need to showthat Theorem 3 holds with some 119904 = 3

By Lemma 17 we obtain that there exists 120585(119905) with 119905 isin

[0 119879) such that

119898(119905) = 119906119909(119905 120585 (119905)) = inf

119909isinR119906119909(119905 119909) for all 119905 isin [0 119879)

(84)

Hence we have 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879)On the other hand since 119902(119905 sdot) R rarr R is

diffeomorphism for any 119905 isin [0 119879) there exists 1199091(119905) isin R

such that 119902(119905 1199091(119905)) = 120585(119905) for all 119905 isin [0 119879) Using the second

equation in system (1) and the trajectory equation (81) wehave

dd119905

120588 (119905 119902 (119905 119909)) = minus119906119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909)) (85)

Hence

120588 (119905 119902 (119905 119909)) = 1205880(119909) 119890

minusint119905

0119906119909(120591119902(120591119909))d120591

(86)

Taking 119909 = 1199091(119905) in (85) together with 119902(119905 119909

1(119905)) = 120585(119905) for

all 119905 isin [0 119879) then

dd119905

120588 (119905 120585 (119905)) = minus119906119909(119905 120585 (119905)) 120588 (119905 120585 (119905)) (87)

Using the assumption 1205880(119909

0) = 0with the point 119909

0defined by

1199060119909(119909

0) = inf

119909isinR1199060119909(119909) in Theorem 3 and letting 120585(0) = 1199090

we deduce that 1205880(120585(0)) = 120588

0(119909

0) = 0 From (87) we get

120588 (119905 120585 (119905)) = 0 for all 119905 isin [0 119879) (88)

If 119892(119909) = (12)119890minus|119909|

119909 isin R then (1 minus 120597119909119909

)minus1119891 = 119892 lowast 119891 for all

119891 isin 1198712(R) so we have 1205972

119909(1 minus 120597

119909119909)minus1119891 = 120597

2

119909119892 lowast119891 = 119892 lowast119891 minus119891

For119898 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= 119875 (119863) [119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

]

+ 1198751(119863) [minus120582119906

2119899+1+ 120573119906

119909119909]

(89)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+ 2119896119906 +

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+ 2119896119906 +

1

2

1205882)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) minus 120573119906

119909

(90)

Note inequality (12)1199062(119909) le 119892 lowast (119906

2+ (12)119906

2

119909)(119909) (see page

347 (58) in [28]) Using Sobolevrsquos embedding theorem and(64) we obtain

1199062(119905 119909) = int

119909

minusinfin

119906119906119909d119909 minus int

+infin

119909

119906119906119909d119909 le int

R

1003816100381610038161003816119906119906

119909

1003816100381610038161003816d119909

le

1

2

int

R

(1199062+ 119906

2

119909) d119909 le

1

2

(10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712)

(91)

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Here we denote 11986212

= (radic22)( 11990602

1198671+ 120578

02

1198712)12 and thus

119906(119905 sdot)119871infin le 119862

12

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining

with (64) yield that 1199061198671 le 119862

11 Togetherwith the definition

of119898(119905) (88) and 119906119909119909

(119905 120585(119905)) = 0 for ae 119905 isin [0 119879) from (90)we derive thatd119898(119905)

d119905le minus

1

2

1198982(119905) minus 120573119898 (119905)

+ [

1199062

2

+ 2119896119906 minus 119892 lowast (2119896119906)

minus120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 120585 (119905))

(92)

Using Youngrsquos inequality we have for all 119905 isin [0 119879)1003817100381710038171003817119892 lowast 119906

1003817100381710038171003817119871infin le

100381710038171003817100381711989210038171003817100381710038171198711119906119871

infin le 11986212

10038161003816100381610038161003816120597119909(119892 lowast 119906

2119899+1)

10038161003816100381610038161003816=

10038161003816100381610038161003816119892119909lowast 119906

2119899+110038161003816100381610038161003816

le1003817100381710038171003817119892119909

10038171003817100381710038171198712

100381710038171003817100381710038171199062119899+110038171003817

1003817100381710038171198712le

1

2

(11986212)2119899+1

1003816100381610038161003816120597119909(119892 lowast 119906)

1003816100381610038161003816le

1003817100381710038171003817119892119909

10038171003817100381710038171198711119906119871

infin le 11986212

(93)

which together with (92) makes us deduce thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870 (94)

where1198981(119905) = 119898 (119905) + 120573

119870 =

1

2(11986212)2+ (4119896 + 120573)119862

12+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(95)

By the assumption of 1198981(0) = 119906

0119909(119909

0) + 120573 lt minusradic2119870 we have

1198982

1(0) gt 2119870 We now claim that 119898

1(119905) lt minusradic2119870 is true for

any 119905 isin [0 119879) In fact assume that 1198981(119905) ge minusradic2119870 for some

119905 isin [0 119879) since 1198981(119905) is continuous we deduce that there

exists 1199050

isin (0 119879) such that 1198982

1(119905) gt 2119870 for 119905 isin [0 119905

0) but

1198982

1(1199050) = 2119870 Combining this with (94) gives d119898

1(119905)d119905 lt 0

ae on [0 1199050) Since 119898

1(119905) is absolutely continuous on [0 119905

0]

so we get the contradiction 1198981(1199050) lt 119898

1(0) = 119906

0119909(119909

0) + 120573 lt

minusradic2119870 Thus we prove the claimNow we deduce that119898

1(119905) is strictly decreasing on [0 119879)

Choosing that 120575 isin (0 1) such that minusradic1205751198981(0) = radic2119870 then

we obtain from (94) thatd119898

1(119905)

d119905le minus

1

2

1198982

1(119905) +

120575

2

1198982

1(0) le minus

1 minus 120575

2

1198982

1(119905)

ae on [0 119879)

(96)

For 1198981(119905) is locally Lipschitz on [0 119879) and strictly negative

we get 11198981(119905) is also locally Lipschitz on [0 119879) This gives

dd119905

(

1

1198981(119905)

) = minus

1

1198982

1(119905)

d1198981(119905)

d119905ge

1 minus 120575

2

ae on [0 119879)

(97)

Integration of this inequality yields

minus

1

1198981(119905)

+

1

1198981(0)

le minus

1 minus 120575

2

119905 ae on [0 119879) (98)

Since 1198981(119905) lt 0 on [0 119879) we obtain that the maximal

existence time 119879 le minus2(1 minus 120575)1198981(0) lt infin Moreover thanks

to1198981(0) = 119906

0119909(119909

0) + 120573 lt 0 again we get that

119906119909(119905 120585 (119905)) le

2 (1199060119909

(1199090) + 120573)

2 + 119905 (1 minus 120575) (1199060119909

(1199090) + 120573)

minus 120573

997888rarr minusinfin(119905 997888rarr

minus2

(1 minus 120575) (1199060119909

(1199090) + 120573)

)

(99)

which completes the proof of Theorem 3

Remark 20 From (81) and (85) one deduces that if (1199060 120588

0) isin

119867119904times 119867

119904minus1 with 119904 gt 32 and the component 119906(119905 119909) does notbreak in finite time 119879 then the component 120588 is uniformlybounded for all (119905 119909) isin [0 119879)timesR In fact if there exists119872 gt 0

such that 119906119909(119905 119909) ge minus119872 for all (119905 119909) isin [0 119879) timesR thus

1003817100381710038171003817120588 (119905)

1003817100381710038171003817119871infin =

1003817100381710038171003817120588 (119905 119902 (119905 sdot))

1003817100381710038171003817119871infin

=

1003817100381710038171003817100381710038171003817

119890minusint119905

0119906119909(120591sdot)d120591

1205880(sdot)

1003817100381710038171003817100381710038171003817119871infin

le 1198901198721198791003817

1003817100381710038171205880

1003817100381710038171003817119871infin

for all 119905 isin [0 119879)

(100)

We find that there is a similar estimation in [25]

Remark 21 Because of the presence of high order term1199062119898

119906119909119909

in system (1) it is difficult to obtain the estimate forthe term minus120573119906

2119898119906119909without the assumption of119898 = 0

43 The Proof of Theorem 4 Similar to the proof ofTheorem 2 here we only need to show that Theorem 4 holdswith 119904 = 3 Note that system (1) with 119896 = 0 is invariant underthe transformation (119906 119909) rarr (minus119906 minus119909) and (120588 119909) rarr (120588 minus119909)Thus if 119906

0is odd and 120588

0is even then the corresponding

solution (119906(119905 119909) 120588(119905 119909)) satisfies that 119906(119905 119909) is odd and 120588(119905 119909)

is even with respect to 119909 for all 119905 isin (0 119879) where 119879 is themaximal existence time Hence 119906(119905 0) = 0 120588

119909(119905 0) = 0

Thanks to the transport equation 120588119905+119906120588

119909+120588119906

119909= 0 at the

point 119909 = 0 we have

dd119905

120588 (119905 0) + 120588 (119905 0) 120597119909119906 (119905 0) = 0

120588 (0 0) = 0

(101)

which derives that 120588(119905 0) = 0 Noting that 1199062119898

119906119909119909

=

120597119909[119906

2119898119906119909] minus 2119898119906

2119898minus1(119906

119909)2 we obtain

120597119909119892 lowast (119906

2119898119906119909119909

)

= 120597119909(1 minus 120597

119909119909)minus1

[120597119909(119906

2119898119906119909) minus 2119898119906

2119898minus11199062

119909]

= 119892 lowast (1199062119898

119906119909) minus 119906

2119898119906119909minus 2119898120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(102)

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

For 119896 = 0 differentiating the first equation in system (7) withrespect to the variable 119909

119906119905+ 119906119906

119909= minus 120597

119909(1 minus 120597

119909119909)minus1

[1199062+

1199062

119909

2

+

1205882

2

]

+ (1 minus 120597119909119909

)minus1

[minus1205821199062119899+1

+ 1205731199062119898

119906119909119909

]

(103)

yields

119906119905119909

= minus

1

2

1199062

119909minus 119906119906

119909119909+ 119906

2+

1

2

1205882

minus 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882) minus 120582120597

119909(119892 lowast 119906

2119899+1)

+ 120573119892 lowast (1199062119898

119906119909) minus 120573119906

2119898119906119909minus 2119898120573120597

119909[119892 lowast (119906

2119898minus11199062

119909)]

(104)Let119872(119905) = 119906

119909(119905 0) and we have

d119872(119905)

d119905le minus

1

2

1198722(119905)

+ [minus120582120597119909(119892 lowast 119906

2119899+1) + 120573119892 lowast (119906

2119898119906119909)

minus2119898120573120597119909(119892 lowast (119906

2119898minus11199062

119909))] (119905 0)

(105)

Denoting 11986211

= ( 11990602

1198671+ 120578

02

1198712)12 and combining with

(64) yield that 1199061198671 le 119862

11 By using Youngrsquos inequality we

deduce for all (119905 119909) isin [0 119879) timesR that10038161003816100381610038161003816119892 lowast (119906

2119898119906119909)

10038161003816100381610038161003816le

100381710038171003817100381711989210038171003817100381710038171198712

100381710038171003817100381710038171199062119898

119906119909

100381710038171003817100381710038171198712

le

1

2

1199062119898

119871infin119906119867

1 =

1

2

1198622119898

1211986211

(106)

10038161003816100381610038161003816120597119909(119892 lowast (119906

2119898minus11199062

119909))

10038161003816100381610038161003816le

1003817100381710038171003817119892119909

1003817100381710038171003817119871infin

100381710038171003817100381710038171199062119898minus1

1199062

119909

100381710038171003817100381710038171198711

le

1

2

1199062119898minus1

119871infin 119906

2

1198671 le

1

2

1198622119898minus1

121198622

11

(107)which together with (93) and (105) derives

d119872(119905)

d119905le minus

1

2

1198722(119905) + 119870

1 (108)

where

1198701=

1

2

120582(11986212)2119899+1

+

1

2

1205731198622119898

1211986211

+ 1205731198981198622119898minus1

121198622

11 (109)

Similarly as in the proof of Theorem 3 from (108) one getsthat119872(119905) lt 119872(0) lt minusradic2119870

1for all 119905 isin [0 119879) and there exists

1205751isin (0 1) such that minusradic120575

1119872(0) = radic2119870

1 Thus

minus

1

119872 (119905)

+

1

119872 (0)

le minus

1 minus 1205751

2

119905 ae on [0 119879) (110)

which implies the maximal existence time 1198791

le minus2(1 minus

1205751)119872(0) lt infin Then

119906119909(119905 0) le

21199060119909

(0)

2 + 119905 (1 minus 1205751) 119906

0119909(0)

997888rarr minusinfin

as 119905 997888rarr minus

2

(1 minus 1205751) 119906

0119909(0)

(111)

which completes the first part of the proof of Theorem 4

Differentiating the transport equation 120588119905+ 119906120588

119909+ 120588119906

119909=

0 with respect to the variable 119909 together with the trajectoryequation (81) we get

d120588119909(119905 119902 (119905 119909))

d119905= minus 119906

119909119909(119905 119902 (119905 119909)) 120588 (119905 119902 (119905 119909))

minus 2119906119909(119905 119902 (119905 119909)) 120588

119909(119905 119902 (119905 119909))

(112)

Taking 119909 = 1199091(119905) and noting that 119902(119905 119909

1(119905)) = 120585(119905) and

119906119909119909

(119905 120585(119905)) = 0 we have

d120588119909(119905 120585 (119905))

d119905= minus2119906

119909(119905 120585 (119905)) 120588

119909(119905 120585 (119905)) (113)

Noting the assumption (84) and 120585(0) = 1199090 120588

0119909(120585(0)) =

1205880119909(119909

0) yields that

120588119909(119905 120585 (119905)) = 120588

0119909(119909

0) 119890

minus2int119905

0119906119909(120591120585(120591))d120591

= 1205880119909

(1199090) 119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

(114)

Thanks to (111) for any 119905 isin [0 119879) we have

119890

minus2int119905

0inf119909isinR

119906119909(120591119909)d120591

ge 119890minus2int119905

0[21199060119909(0)(2+120591(1minus120575

1)1199060119909(0))]d120591

= 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

(115)

Note that 119890(minus4(1minus120575

1)) ln[1+((1minus120575

1)1199060119909(0)2)119905]

rarr +infin as 119905 rarr

minus2(1 minus 1205751)119906

0119909(0)

Therefore if 1205880119909(119909

0) gt 0 from (115) for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) we have

sup119909isinR

120588119909(119905 119909) ge 120588

119909(119905 120585 (119905)) 997888rarr +infin as 119905 997888rarr 119879

minus

2

(116)

On the other hand if 1205880119909(119909

0) lt 0 for 0 lt 119879

2le minus2(1 minus

1205751)119906

0119909(0) it follows from (115) that

inf119909isinR

120588119909(119905 119909) le 120588

119909(119905 120585 (119905)) 997888rarr minusinfin as 119905 997888rarr 119879

minus

2

(117)

Now we complete the proof of Theorem 4

Remark 22 Under the same conditions in Theorem 3 with119904 gt 52 and 120588

0119909(119909

0) = 0 we follow a similar argument of

Theorem 4 to obtain the same blow-up results (i) and (ii) inTheorem 4

Remark 23 Thanks to the assumption of 119896 = 0 system (1) isinvariant under the transformation (119906 119909) rarr (minus119906 minus119909) and(120588 119909) rarr (120588 minus119909)

44 Blow-Up Rate Having established the wave-breakingresults for system (1) we give the estimate for the blow-uprate of solutions to system (1)

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

Proof of Theorem 5 Noting that |119892 lowast 120578| le (12)( 1198922

1198712+

1205782

1198712) le (12)(1+ 119906

02

1198671+ 120578

02

1198712) we have

0 le 119892 lowast (1199062+

1

2

1199062

119909+

1

2

1205882)

le10038171003817100381710038171198921003817100381710038171003817119871infin

1003817100381710038171003817100381710038171003817

1199062+

1

2

1199062

119909+

1

2

120578210038171003817100381710038171003817100381710038171198711

+1003816100381610038161003816119892 lowast 120578

1003816100381610038161003816+

1

2

le 1 +10038171003817100381710038171199060

1003817100381710038171003817

2

1198671 +

10038171003817100381710038171205780

1003817100381710038171003817

2

1198712 = 1 + 2119862

2

12

(118)

Similar to the proof of Theorem 3 we obtaind119898

1(119905)

d119905le minus

1

2

1198982

1(119905) + 119870

2 (119)

where

1198702= 1 + 3(119862

12)2

+ (4119896 + 120573)11986212

+

1

2

120582(11986212)2119899+1

+

1

2

1205732

(120)

Therefore choosing 120576 isin (0 12) and using (99) we find1199050

isin (0 119879) such that 1198981(1199050) lt minusradic2119870

2+ 119870

2120576 Since

1198981(119905) is locally Lipschitz it follows that 119898

1(119905) is absolutely

continuous Integrating (119) on interval [1199050 119905) with 119905 isin

[1199050 119879) we deduce that119898

1(119905) is decreasing on [119905

0 119879) and

1198981(119905) lt minusradic2119870

2+

1198702

120576

lt minusradic1198702

120576

119905 isin [1199050 119879) (121)

It deduces from (119) that10038161003816100381610038161003816100381610038161003816

d1198981(119905)

d119905+

1

2

1198982

1(119905)

10038161003816100381610038161003816100381610038161003816

le 1198702997904rArr minus119870

2minus

1

2

1198982

1(119905)

le

d1198981(119905)

d119905le 119870

2minus

1

2

1198982

1(119905)

ae 119905 isin (1199050 119879)

(122)

Noting that1198981(119905) lt 0 on (119905

0 119879) we obtain

1

2

minus 120576 le

dd119905

(

1

1198981(119905)

) le

1

2

+ 120576 for ae 119905 isin (1199050 119879) (123)

Integrating the above relation on (119905 119879) with 119905 isin [1199050 119879) and

noting that

lim119905rarr119879

minus

1198981(119905) = minusinfin (124)

we have ((12) minus 120576)(119879 minus 119905) le minus(11198981(119905)) le ((12) + 120576)(119879 minus 119905)

Since 120576 isin (0 12) is arbitrary it deduces from (95) and(124) that

lim119905rarr119879

minus

[inf119909isinR

(119906119909(119905 119909) + 120573) (119879 minus 119905)] = minus2 (125)

which completes the proof of Theorem 5

Remark 24 Let 119879 lt infin be the blow-up time of thecorresponding solution to system (1)The initial data (119906

0 120588

0minus

1) isin 119867119904times 119867

119904minus1 with 119904 gt 32 satisfies the assumptions ofTheorem 4 Then

lim119905rarr119879

minus

[inf119909isinR

119906119909(119905 0) (119879 minus 119905)] = minus2 (126)

Proof The argument is similar to the proof of Theorem 5here we omit it

5 The Proof of Theorem 6

In this section we first give a lemma on global strong solu-tions to system (1) Then we present the proof of Theorem 6We need to pay more attention to the assumption 119898 = 0 and1205880(119909) = 0 for all 119909 isin R

Lemma 25 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 32 Let T gt 0 be the maximal existencetime of corresponding solution (119906 120588) to system (1) in the case of119898 = 0 with initial data (119906

0 120588

0) If 120588

0(119909) = 0 for all 119909 isin R then

there exists a constant 1205732gt 0 such that

lim119905rarr119879

minus

inf119909isinR

119906119909(119905 119909)

ge minus

1

21205732

[3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879minus 120573

(127)

where1198704is a fixed constant defined in (132)

Proof From Lemma 18 we have inf119909isinR119906119909(119905 119902(119905 119909)) =

inf119909isinR119906119909(119905 119909) for all 119905 isin [0 119879) Let 119867(119905 119909) = 119906

119909(119905 119902(119905 119909)) +

120573 120574(119905 119909) = 120588(119905 119902(119905 119909)) and then

d119867(119905 119909)

d119905= (119906

119905119909+ 119906119906

119909119909) (119905 119902 (119905 119909))

d120574 (119905 119909)

d119905= minus120574 (119905 119909) [119867 (119905 119909) minus 120573]

(128)

Noting119898 = 0 and using (90) yield

d119867(119905 119909)

d119905= minus

1

2

1198672+

1

2

1205742+ 119891 (119905 119909) +

1

2

1205732 (129)

where

119891 (119905 119909) = [1199062+ 2119896119906 minus 119892 lowast (119906

2+

1199062

119909

2

+ 2119896119906 +

1205882

2

)

minus 120582120597119909(119892 lowast 119906

2119899+1) + 120573120597

119909(119892 lowast 119906) ] (119905 119902)

(130)

From the proof of Theorems 3 and 5 one deduces that

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816le 3119862

2

12+ 1 + (4119896 + 120573)119862

12

+

1

2

120582(11986212)2119899+1

= 1198703

for all (119905 119909) isin [0 119879) timesR

(131)

For convenience here we denote

1198704= max(119870

3+

1

2

120573) (132)

From (85) we obtain that 120574(119905 119909) has the same sign with120574(0 119909) = 120588

0(119909) for all 119909 isin R For 120578

0(119909) isin 119867

119904minus1 with 119904 gt 32using Sobolevrsquos embedding theorem we have 120578

0(119909) isin 119862

119888(R)

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 15

and there exists a constant 1205731such that |120578

0(119909)| le 12 for all

|119909| ge 1205731 Since 120578

0(119909) isin 119862

119888(R) and 120588

0(119909) = 0 for all 119909 isin R it

follows thatinf|119909|le1205731

1003816100381610038161003816120574 (0 119909)

1003816100381610038161003816= inf

|119909|le1205731

10038161003816100381610038161205880(119909)

1003816100381610038161003816gt 0 (133)

Taking 1205732= min(12 inf

|119909|le1205731

|120574(0 119909)|) then |120574(0 119909)| ge 1205732gt

0 for all 119909 isin R and

120574 (119905 119909) 120574 (0 119909) gt 0 for all 119909 isin R (134)

Now we consider the following Lyapunov function

119908 (119905 119909) = 120574 (0 119909) 120574 (119905 119909) +

120574 (0 119909)

120574 (119905 119909)

[1 + 1198672(119905 119909)]

(119905 119909) isin [0 119879) timesR

(135)

Applying Sobolevrsquos embedding theorem yields

119908 (0 119909) = 1205742(0 119909) + 1 + 119867

2(0 119909)

le 3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1

(136)

Differentiating (135) with respect to 119905 and using (129) and(132) we obtain

d119908 (119905 119909)

d119905= 2

120574 (0 119909)

120574 (119905 119909)

119867 (119905 119909) [

1

2

+ 119891 (119905 119909)]

+

120574 (0 119909)

(120574 (119905 119909))

(1205742minus 1) 120573

le (1198703+

1

2

)

(120574 (0 119909))

120574 (119905 119909)

[1 + 1198672(119905 119909)]

+ 120573120574 (0 119909) 120574 (119905 119909)

le 1198704119908 (119905 119909)

(137)

By using Gronwallrsquos inequality and (136) we deduce that

119908 (119905 119909) le 119908 (0 119909) 1198901198704119905

le (3 + 21205732+ 2

10038171003817100381710038171199060 120588

0minus 1

1003817100381710038171003817

2

119867119904times119867119904minus1) 119890

1198704119879

(138)

for all (119905 119909) isin [0 119879) times R On the other hand from (135) wehave

119908 (119905 119909) ge 2radic1205742(0 119909) [1 + 119867

2(119905 119909)]

ge 21205732

|119867 (119905 119909)|

for all (119905 119909) isin [0 119879) timesR

(139)

which together with (138) yields that for all (119905 119909) isin [0 119879)timesR

119867(119905 119909) ge minus

1

21205732

119908 (119905 119909) ge minus

1

21205732

times [3 + 21205732+ 2

1003817100381710038171003817(119906

0 120588

0minus 1)

1003817100381710038171003817

2

119867119904times119867119904minus1] 119890

1198704119879

(140)

Then by the definition of 119867(119905 119909) we complete the proof ofLemma 25

Proof of Theorem 6 Combining the results ofTheorem 1 (72)and Lemma 25 we complete the proof of Theorem 6

Remark 26 Assume 119898 = 0 in system (1) and (1199060 120588

0minus 1) isin

119867119904times 119867

119904minus1 with 119904 gt 52 and 1205880(119909) = 0 for all 119909 isin R Let 119879

be the maximal existence time of the corresponding solution(119906 120588minus1) to system (1)with119898 = 0 and then the correspondingsolution blows up in finite time if and only if

lim119905rarr119879

sup 1003817100381710038171003817120588119909(119905 sdot)

1003817100381710038171003817119871infin = +infin (141)

Remark 27 For the same difficulty stated in Remark 21 herewe only obtain the global existence result for solutions tosystem (1) with the assumption119898 = 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for anumber of valuable comments and suggestions This paper issupported by National Natural Science Foundation of China(71003082) and Fundamental Research Funds for the CentralUniversities (SWJTU12CX061 and SWJTU09ZT36)

References

[1] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[2] A Constantin and D Lannes ldquoThe hydrodynamical rele-vance of theCamassa-HolmandDegasperis-Procesi equationsrdquoArchive for Rational Mechanics and Analysis vol 192 no 1 pp165ndash186 2009

[3] A Constantin ldquoThe trajectories of particles in Stokes wavesrdquoInventiones Mathematicae vol 166 no 3 pp 523ndash535 2006

[4] A Constantin and J Escher ldquoParticle trajectories in solitarywaterwavesrdquoBulletin of the AmericanMathematical Society vol44 no 3 pp 423ndash431 2007

[5] A Constantin and W A Strauss ldquoStability of peakonsrdquo Com-munications on Pure and AppliedMathematics vol 53 no 5 pp603ndash610 2000

[6] A Constantin and W A Strauss ldquoStability of the Camassa-Holm solitonsrdquo Journal of Nonlinear Science vol 12 no 4 pp415ndash422 2002

[7] A Constantin and W A Strauss ldquoStability of a class of solitarywaves in compressible elastic rodsrdquo Physics Letters A vol 270no 3-4 pp 140ndash148 2000

[8] Y A Li and P J Olver ldquoWell-posedness and blow-up solutionsfor an integrable nonlinearly dispersive model wave equationrdquoJournal of Differential Equations vol 162 no 1 pp 27ndash63 2000

[9] S Lai and Y Wu ldquoLocal well-posedness and weak solutions fora weakly dissipative Camassa-Holm equationrdquo Scientia SinicaMathematica vol 40 no 9 pp 901ndash920 2010

[10] S Lai and Y Wu ldquoThe local well-posedness and existenceof weak solutions for a generalized Camassa-Holm equationrdquo

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Mathematical Problems in Engineering

Journal of Differential Equations vol 248 no 8 pp 2038ndash20632010

[11] S Lai and Y Wu ldquoGlobal solutions and blow-up phenomena toa shallow water equationrdquo Journal of Differential Equations vol249 no 3 pp 693ndash706 2010

[12] S Lai and Y Wu ldquoA model containing both the Camassa-Holmand Degasperis-Procesi equationsrdquo Journal of MathematicalAnalysis and Applications vol 374 no 2 pp 458ndash469 2011

[13] S Lai andYWu ldquoThe study of globalweak solutions for a gener-alized hyperelastic-rod wave equationrdquo Nonlinear Analysis vol80 pp 96ndash108 2013

[14] C Mu S Zhou and R Zeng ldquoWell-posedness and blow-upphenomena for a higher order shallow water equationrdquo Journalof Differential Equations vol 251 no 12 pp 3488ndash3499 2011

[15] L Tian P Zhang and L Xia ldquoGlobal existence for the higher-order Camassa-Holm shallow water equationrdquoNonlinear Anal-ysis vol 74 no 7 pp 2468ndash2474 2011

[16] D-X Kong ldquoGlobal structure instability of Riemann solutionsof quasilinear hyperbolic systems of conservation laws rarefac-tion wavesrdquo Journal of Differential Equations vol 219 no 2 pp421ndash450 2005

[17] Y Guo S Lai and Y Wu ldquoOn existence and uniqueness ofthe global weak solution for a shallow water equationrdquo AppliedMathematics and Computation vol 218 no 23 pp 11410ndash114202012

[18] Y Mi and C Mu ldquoWell-posedness and analyticity for theCauchy problem for the generalized Camassa-Holm equationrdquoJournal of Mathematical Analysis and Applications vol 405 no1 pp 173ndash182 2013

[19] S Wu and Z Yin ldquoGlobal existence and blow-up phenomenafor the weakly dissipative Camassa-Holm equationrdquo Journal ofDifferential Equations vol 246 no 11 pp 4309ndash4321 2009

[20] A Constantin and R I Ivanov ldquoOn an integrable 2-componentCamassa-Holm shallow water systemrdquo Physics Letters A vol372 no 48 pp 7129ndash7132 2008

[21] K Yan and Z Yin ldquoOn the Cauchy problem for a 2-componentDegasperis-Procesi systemrdquo Journal of Differential Equationsvol 252 no 3 pp 2131ndash2159 2012

[22] K Yan and Z Yin ldquoAnalytic solutions of the Cauchy prob-lem for 2-component shallow water systemsrdquo MathematischeZeitschrift vol 269 no 3-4 pp 1113ndash1127 2011

[23] C Guan and Z Yin ldquoGlobal existence and blow-up phenomenafor an integrable 2-component Camassa-Holm shallow watersystemrdquo Journal of Differential Equations vol 248 no 8 pp2003ndash2014 2010

[24] C Guan and Z Yin ldquoGlobal weak solutions for a 2-componentCamassa-Holm shallow water systemrdquo Journal of FunctionalAnalysis vol 260 no 4 pp 1132ndash1154 2011

[25] J EscherO Lechtenfeld andZ Yin ldquoWell-posedness and blow-up phenomena for the 2-component Camassa-Holm equationrdquoDiscrete and Continuous Dynamical Systems vol 19 no 3 pp493ndash513 2007

[26] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the 2-component Camassa-Holm systemrdquo Journal ofFunctional Analysis vol 258 no 12 pp 4251ndash4278 2010

[27] G Gui and Y Liu ldquoOn the Cauchy problem for the 2-component camassa-Holm systemrdquo Mathematische Zeitschriftvol 268 no 1-2 pp 45ndash46 2011

[28] M Chen S-Q Liu and Y Zhang ldquoA 2-component generaliza-tion of the Camassa-Holm equation and its solutionsrdquo Letters inMathematical Physics vol 75 no 1 pp 1ndash15 2006

[29] M Zhu ldquoBlow-up global existence and persistence proper-ties for the coupled Camassa-Holm equationsrdquo MathematicalPhysics Analysis and Geometry vol 14 no 3 pp 197ndash209 2011

[30] Z Guo and L Ni ldquoPersistence properties and unique continu-ation of solutions to a 2-component Camassa-Holm equationrdquoMathematical Physics Analysis and Geometry vol 14 no 2 pp101ndash114 2011

[31] Y Fu and C Qu ldquoUnique continuation and persistence prop-erties of solutions of the 2-component Degasperis-Procesiequationsrdquo Acta Mathematica Scientia vol 32 no 2 pp 652ndash662 2012

[32] Y Jin and Z Jiang ldquoWave breaking of an integrable Camassa-Holm system with 2-componentsrdquo Nonlinear Analysis vol 95pp 107ndash116 2014

[33] A Shabat and LM Alonso ldquoOn the prolongation of a hierarchyof hydrodynamic chainsrdquo in New Trends in Integrability andPartial Solvability vol 132 of NATO Science Series II pp 263ndash280 Springer Dordrecht The Netherlands 2004

[34] L Tian W Yan and G Gui ldquoOn the local well posednessand blow-up solution of a coupled Camassa-Holm equations inBesov spacesrdquo Journal of Mathematical Physics vol 53 ArticleID 013701 2012

[35] W Chen X Deng and J Zhang ldquoBlow up and blow-up ratefor the generalized 2-component Camassa-Holm equationrdquoInternational Journal of Nonlinear Science vol 12 no 3 pp 313ndash322 2011

[36] W Chen L Tian X Deng and J Zhang ldquoWave breaking fora generalized weakly dissipative 2-component Camassa-Holmsystemrdquo Journal of Mathematical Analysis and Applications vol400 no 2 pp 406ndash417 2013

[37] C Qu Y Fu and Y Liu ldquoWell-posedness wave breaking andpeakons for a modified 120583-Camassa-Holm equationrdquo Journal ofFunctional Analysis vol 266 no 2 pp 433ndash477 2013

[38] Y Fu G Gui Y Liu and C Qu ldquoOn the Cauchy problemfor the integrable modified Camassa-Holm equation with cubicnonlinearityrdquo Journal of Differential Equations vol 255 no 7pp 1905ndash1938 2013

[39] R Danchin ldquoA few remarks on the Camassa-Holm equationrdquoDifferential and Integral Equations vol 14 no 8 pp 953ndash9882001

[40] R Danchin Fourier Analysis Methods for PDEs Lecture Notes2005

[41] W Yan Y Li and Y Zhang ldquoThe Cauchy problem for theintegrable Novikov equationrdquo Journal of Differential Equationsvol 253 no 1 pp 298ndash318 2012

[42] H Bahouri J-Y Chemin and R Danchin Fourier Analysis andNonlinear Partial Differential Equations vol 343 ofGrundlehrender Mathematischen Wissenschafte Springer Berlin Germany2011

[43] Z Yin ldquoOn the Cauchy problem for an integrable equation withpeakon solutionsrdquo Illinois Journal of Mathematics vol 47 no 3pp 649ndash666 2003

[44] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998

[45] A Constantin ldquoExistence of permanent and breaking waves fora shallow water equation a geometric approachrdquo Annales delrsquoInstitut Fourier vol 50 no 2 pp 321ndash362 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of