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وزارة التعليم العايل والبحث العلمي
BADJI MOKHTAR UNIVERSITY OF ANNABA UNIVERSITÉ BADJI MOKHTAR DE ANNABA
جامعة باجي مختار -عنابة -
Faculté des Sciences
Département de Mathématiques
MÉMOIRE
Présenté en vue de l’obtention du diplôme de
MAGISTER EN MATHÉMATIQUES ÉCOLE DOCTORALE EN MATHÉMATIQUES
Par
Khaled ZENNIR
Intitulé
Existence et Comportement Asymptotique des
Solutions d’une Equation de Viscoélasticité
Non Linéaire de type Hyperbolique
Dirigé par
Prof. Hocine SISSAOUI
Option Systèmes Dynamiques et Analyse Fonctionnelle
Devant le jury
PRÉSIDENT
B. KHODJA
PROF.
U. B. M. ANNABA
RAPPORTEUR
H. SISSAOUI
PROF.
U. B. M. ANNABA
ÉXAMINATEUR
L. NISSE
M. C.
U. B. M. ANNABA
ÉXAMINATEUR
Y. LASKRI
M. C.
U. B. M. ANNABA
INVITE
B. SAID-HOUARI
Dr.
U. B. M. ANNABA
Année 2008
وزارة التعليم العايل والبحث العلمي
BADJI MOKHTAR UNIVERSITY OF ANNABA UNIVERSITÉ BADJI MOKHTAR DE ANNABA
جامعة باجي مختار -عنابة -
Sciences Faculty
Department of Mathematics
THESIS
Submitted for the obtention of
MAGISTER DIPLOMA IN MATHEMATICS DOCTORAL SCHOOL IN MATHEMATICS
By
Khaled ZENNIR
Existence et Comportement Asymptotique des
Solutions d’une Equation de Viscoélasticité
Non Linéaire de type Hyperbolique
Directed by
First Advisor: Prof. Hocine SISSAOUI Second Advisor : Dr. Belkacem SAÏD-HOUARI
Option
Dynamic Systems and Functional Analysis
PRESIDENT
B. KHODJA
PROF.
U. B. M. ANNABA
SUPERVISOR
H. SISSAOUI
PROF.
U. B. M. ANNABA
EXAMINER
L. NISSE
M. C.
U. B. M. ANNABA
EXAMINER
Y. LASKRI
M. C.
U. B. M. ANNABA
2008
Existence and Asymptotic Behavior of
Solutions of a Non Linear Viscoelastic
Hyperbolic Equation
ThanksThanksThanksThanks
In the first place at all, my gratitude goes to Prof. H. SISSAOUI, my supervisor. He always listens, advices, encourages and guides me, he was behind this work. I shall always be impressed by his availability and the care with which he comes to read and correct my manuscript.
My warm thanks go to Dr.B.SAID-HOUARI, who supervised this thesis with great patience and kindness. He was able to motivate each step of my work by relevant remarks and let me able to advance in my research, beginning by the DES study in university of Skikda. I thank him very sincerely for his availability even at a distance, with him I have a lot of fun to work.
I wish to thank Prof B. KHODJA, who makes me the honour of chairing the jury of my defense.
I am very grateful to Dr. L. NISSE and Dr. Y. LASKRI to have agreed to rescind this thesis and to be part of my defense panel.
Of course, this thesis is the culmination of many years of study. I am indebted to all those who guided me on the road of mathematics.
I thank from my heart, my teachers whose encouraged me during my studies and have distracted me the mathematics.
I think for my parents, whose the work could not succeed without their endless support and encouragement. They find in the realization of this work, the culmination of their efforts and express my most affectionate gratitude.
Khaled ZENNIR
Contents
1 Preliminary 1
1.1 Banach Spaces - De�nition and Properties . . . . . . . . . . . . . . . . . . . . 2
1.1.1 The weak and weak star topologies . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The Lp () spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 The Sobolev space Wm;p() . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 The Lp (0; T;X) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 Some Algebraic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Existence Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Gronwell�s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 The mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Local Existence 17
2.1 Local Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Global Existence and Energy Decay 39
3.1 Global Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Decay of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Exponential Growth 59
4.1 Growth result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
i
ii
Abstract
Our work, in this thesis, lies in the study, under some conditions on p, m and the functional g, the
existence and asymptotic behavior of solutions of a nonlinear viscoelastic hyperbolic problem
of the form 8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
utt ��u� !�ut +tR0
g(t� s)�u(s; x)ds
+a jutjm�2 ut = b jujp�2 u; x 2 ; t > 0
u (0; x) = u0 (x) ; x 2 ut (0; x) = u1 (x) ; x 2
u (t; x) = 0; x 2 �; t > 0
; (P)
where is a bounded domain in RN (N � 1), with smooth boundary �; a; b; w are positiveconstants, m � 2; p � 2; and the function g satisfying some appropriate conditions.
Our results contain and generalize some existing results in literature. To prove our results many
theorems were introduced.
Keywords: Nonlinear damping, strong damping, viscoelasticity, nonlinear source, local solutions,
global solutions, exponential decay, polynomial decay, growth.
iii
Résumé
Notre travail, dans ce memoire consiste à étudier l�éxistence et le comportement asymptotique des
solutions d�un problème de viscoelasticité non lineaire de type hyperbolique suivant:8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
utt ��u� !�ut +tR0
g(t� s)�u(s; x)ds
+a jutjm�2 ut = b jujp�2 u; x 2 ; t > 0
u (0; x) = u0 (x) ; x 2 ut (0; x) = u1 (x) ; x 2
u (t; x) = 0; x 2 �; t > 0
; (P)
où, est un domaine borné de RN (N � 1), avec frontière assez regulière �; a; b; w sont desconstantes positives, m � 2; p � 2; et la fonction g satisfaite quelques conditions.
Nos résultats contiennent et généralisent certains résultats d�existences dans la littérature.
Pour la preuve, beaucoup théorèmes ont été présentés
Mots clés: Dissipation nonlinéaire, viscoelasticité, source nonlinéaire, solutions locale, solutions
globale, décroissance exponentielle de l�énergie, décroissance polynomiale, croissement.
iv
Notations
: bounded domain in RN :� : topological boundary of :
x = (x1; x2; :::; xN) : generic point of RN :dx = dx1dx2:::dxN : Lebesgue measuring on :
ru : gradient of u:�u : Laplacien of u:
f+; f� : max(f; 0); max(�f; 0):a:e : almost everywhere.
p0 : conjugate of p; i:e1
p+1
p0= 1:
D() : space of di¤erentiable functions with compact support in :
D0() : distribution space.
Ck () : space of functions k�times continuously di¤erentiable in :C0 () : space of continuous functions null board in :
Lp () : Space of functions p�th power integrated on with measure of dx:
kfkp =�R
jf(x)jp�1p
:
W 1;p () =nu 2 Lp () ; ru 2 (Lp ())N
o:
kuk1;p =�kukpp + kruk
pp
�1p:
W 1;p0 () : the closure of D () in W 1;p () :
W�1;p0 () : the dual space of W 1;p0 () :
H : Hilbert space.
H10 = W 1;2
0 :
If X is a Banach space
Lp (0; T ;X) =
�f : (0; T )! X is measurable ;
TR0
kf(t)kpX dt <1�:
L1 (0; T ;X) =
(f : (0; T )! X is measurable ;ess� sup
t2(0;T )kf(t)kpX <1
):
Ck ([0; T ] ;X) : Space of functions k�times continuously di¤erentiable for [0; T ]! X:
D ([0; T ] ;X) : space of functions continuously di¤erentiable with compact support in [0; T ] :
BX = fx 2 X; kxk � 1g : unit ball.
v
Introduction
In this thesis we consider the following nonlinear viscoelastic hyperbolic problem8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:
utt ��u� !�ut +tR0
g(t� s)�u(s; x)ds
+a jutjm�2 ut = b jujp�2 u; x 2 ; t > 0
u (0; x) = u0 (x) ; ut (0; x) = u1 (x) ; x 2
u (t; x) = 0; x 2 �; t > 0
; (1)
where is a bounded domain inRN (N � 1), with smooth boundary �; a; b; w are positive constants,and m � 2; p � 2: The function g(t) is assumed to be a positive nonincreasing function de�ned onR+ and satis�es the following conditions:(G1): g : R+ �! R+ is a bounded C1-function such that
g(0) > 0; 1�1Z0
g(s)ds = l > 0:
(G2): g(t) � 0; g0(t) � 0; g(t) � ��g0(t); 8 t � 0; � > 0:
In the physical point of view, this type of problems arise usually in viscoelasticity. This type
of problems have been considered �rst by Dafermos [12], in 1970, where the general decay was
discussed. A related problems to (1) have attracted a great deal of attention in the last two
decades, and many results have been appeared on the existence and long time behavior of solutions.
See in this directions [2;3;5� 8;18;29;33;34;38] and references therein.
In the absence of the strong damping �ut; that is for w = 0; and when the function g vanishes
identically ( i.e: g = 0); then problem (1) reduced to the following initial boundary damped wave
equation with nonlinear damping and nonlinear sources terms.
utt ��u+ a jutjm�2 ut = b jujp�2 u: (2)
Some special cases of equation (2) arise in quantum �eld theory which describe the motion of
charged mesons in an electromagnetic �eld.
Equation (2) together with initial and boundary conditions of Dirichlet type, has been extensively
studied and results concerning existence, blow up and asymptotic behavior of smooth, as well
vi
as weak solutions have been established by several authors over the past three decades. Some
interesting results have been summarized by Said-Houari in his master thesis [42]:
For b = 0, that is in the absence of the source term, it is well known that the damping term
a jutjm�2 ut assures global existence and decay of the solution energy for arbitrary initial data ( seefor instance [17] and [21]):
For a = 0; the source term causes �nite-time blow-up of solutions with a large initial data ( negative
initial energy). That is to say, the norm of our solution u(t; x) in the energy space reaches +1when the time t approaches certain value T � called " the blow up time", ( see [1] and [20] for more
details).
The interaction between the damping term a jutjm�2 ut and the source term b jujp�2 u makes theproblem more interesting. This situation was �rst considered by Levine [23;24] in the linear
damping case (m = 2), where he showed that solutions with negative initial energy blow up in �nite
time T �. The main ingredient used in [23] and [24] is the " concavity method" where the basic
idea of this method is to construct a positive function L(t) of the solution and show that for some
> 0; the function L� (t) is a positive concave function of t: In order to �nd such ; it su¢ ces to
verify that:d2L� (t)
dt2= � L� �2(t)
�LL00 � (1 + )L02(t)
�� 0; 8t � 0:
This is equivalent to prove that L(t) satis�es the di¤erential inequality
LL00 � (1 + )L02(t) � 0; 8t � 0:
Unfortunately, this method fails in the case of nonlinear damping term (m > 2):
Georgiev and Todorova in their famous paper [14], extended Levine�s result to the nonlinear damp-
ing case (m > 2): More precisely, in [14] and by combining the Galerkin approximation with the
contraction mapping theorem, the authors showed that problem (2) in a bounded domain with
initial and boundary conditions of Dirichlet type has a unique solution in the interval [0; T ) provided
that T is small enough. Also, they proved that the obtained solutions continue to exist globally in
time if m � p and the initial data are small enough. Whereas for p > m the unique solution of
problem (2) blows up in �nite time provided that the initial data are large enough. ( i.e: the initial
energy is su¢ ciently negative).
This later result has been pushed by Messaoudi in [35] to the situation where the initial energy
E(0) < 0: For more general result in this direction, we refer the interested reader to the works of
Vitillaro [47]; Levine [25] and Serrin and Messaoudi and Said-Houari [32]:
vii
In the presence of the viscoelastic term (g 6= 0) and for w = 0; our problem (1) becomes8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:
utt ��u+tR0
g(t� s)�u(s; x)ds
+a jutjm�2 ut = b jujp�2 u; x 2 ; t > 0
u (0; x) = u0 (x) ; ut (0; x) = u1 (x) ; x 2
u (t; x) = 0; x 2 �; t > 0
: (3)
For a = 0; problem (3) has been investigated by Berrimi and Messaoudi [3]: They established the
local existence result by using the Galerkin method together with the contraction mapping theorem.
Also, they showed that for a suitable initial data, then the local solution is global in time and in
addition, they showed that the dissipation given by the viscoelastic integral term is strong enough to
stabilize the oscillations of the solution with the same rate of decaying ( exponential or polynomial)
of the kernel g: Also their result has been obtained under weaker conditions than those used by
Cavalcanti et al [7]; in which a similar problem has been addressed.
Messaoudi in [29]; showed that under appropriate conditions between m; p and g a blow up and
global existence result, of course his work generalizes the results by Georgiev and Todorova [14]
and Messaoudi [29]:
One of the main direction of the research in this �eld seems to �nd the minimal dissipation such
that the solutions of problems similar to (3) decay uniformly to zero, as time goes to in�nity.
Consequently, several authors introduced di¤erent types of dissipative mechanisms to stabilize these
problems. For example, a localized frictional linear damping of the form a(x)ut acting in sub-domain
w � has been considered by Cavalcanti et al [7]: More precisely the authors in [6] looked into
the following problem
utt ��u+tZ0
g(t� s)�u(s; x)ds+ a(x)ut + juj u = 0: (4)
for > 0; g a positive function and a : ! R+ a function, which may be null on a part of thedomain :
By assuming a(x) � a0 > 0 on the sub-domain w � ; the authors showed a decay result of an
exponential rate, provided that the kernel g satis�es
� �1g(t) � g0(t) � ��2g(t); t � 0; (5)
viii
and kgkL1(0;1) is small enough.
This later result has been improved by Berrimi and Messaoudi [2], in which they showed that the
viscoelastic dissipation alone is strong enough to stabilize the problem even with an exponential
rate.
In many existing works on this �eld, the following conditions on the kernel
g0(t) � ��gp(t); t � 0; p � 1; (6)
is crucial in the proof of the stability.
For a viscoelastic systems with oscillating kernels, we mention the work by Rivera et al [36]. In
that work the authors proved that if the kernel satis�es g(0) > 0 and decays exponentially to zero,
that is for p = 1 in (6), then the solution also decays exponentially to zero. On the other hand,
if the kernel decays polynomially, i.e. (p > 1) in the inequality (6), then the solution also decays
polynomially with the same rate of decay.
In the presence of the strong damping (w > 0) and in the absence of the viscoelastic term (g = 0),
the problem (1) takes the following form8>>>>>>><>>>>>>>:
utt ��u� !�ut + a jutjm�2 ut = b jujp�2 u; x 2 ; t > 0
u (0; x) = u0 (x) ; ut (0; x) = u1 (x) ; x 2
u (t; x) = 0; x 2 �; t > 0
: (7)
Problem (7) represents the wave equation with a strong damping �ut. When m = 2, this problem
has been studied by Gazzola and Squassina [13]: In their work, the authors proved some results
on well posedness and asymptotic behavior of solutions. They showed the global existence and
polynomial decay property of solutions provided that the initial data is in the potential well.
The proof in [13] is based on a method used in [19]. Unfortunately their decay rate is not opti-
mal, and their result has been improved by Gerbi and Said-Houari [16], by using an appropriate
modi�cation of the energy method and some di¤erential and integral inequalities.
Introducing a strong damping term �ut makes the problem from that considered in [42] and [14],
for this reason less results where known for the wave equation with strong damping and many
problem remain unsolved. ( See [13] and the recent work by Gerbi and Said-Houari [15]):
In this thesis, we investigated problem (1), in which all the damping mechanism have been considered
in the same time ( i.e. w > 0; g 6= 0; and m � 2); these assumptions make our problem di¤erent
ix
form those studied in the literature, specially the blow up result / exponential growth of solutions
(chapter4).
This thesis is organized as follows:
Chapter1:
In this chapter we introduce some notation and prepare some material needed for our work. The
main results of this chapter such as: the Lp� spaces, the Sobolev spaces, di¤erential and integral
inequalities and other theorems of functional analysis, can found in the books [4] and [43]:
Chapter2:
This chapter is devoted to the study of the local existence result, the main ingredient used in this
chapter is the Galerkin approximations ( the compactness method) introduced in the book of Lions
[26]; together with the �x point method.
Indeed, we consider �rst for u 2 C ([0; T ] ; H10 ) given, the following problem
vtt ��v � !�vt +
tZ0
g(t� s)�v(s; x)ds+ a jvtjm�2 vt = b jujp�2 u; x 2 ; t > 0 (8)
with the initial data
v (0; x) = u0 (x) ; vt (0; x) = u1 (x) ; x 2 (9)
and boundary conditions of the form
v (t; x) = 0; x 2 �; t > 0; (10)
and we will show that problem (8) � (10) has a unique local solutions v by the Faedo-Galerkinmethod, which consists in constructing approximations of the solution, then we obtain a priori
estimates necessary to guarantee the convergence of these approximations. We recall here that the
presence of nonlinearity on the damping term a jvtjm�2 vt forces us to go until the second a prioriestimate. We point out that the contraction semigroup method fails here, because of the presence
of the nonlinear terms.
Once the local solution v exists, we will use the contraction mapping theorem to show the local
existence of our problem (1). This will be done under the assumption that T is required to be small
enough (see formula (2:77)):
Chapter 3:
Our main purpose in this chapter is tow-fold:
First, we introduce a set W de�ned in (3:5) called " the potential well" or " stable set" and we
show that if we restrict our initial data in this set, then our solution obtained in chapter 2 is global
in time, that is to say, the norm
kutk2 + kruk2 ;
x
in the energy space L2() � H10 () of our solution is bounded by a constant independent of the
time t:
Second, We show that, if our solution is global in time, ( i.e: by assuming that the initial data
u0 2 W ) and if our function g satis�es the condition (6) ( for p = 1); then our solution decays timeasymptotically to zero. More precisely we prove that the decay rate is of the form (1 + t)2=(2�m)
if m > 2; whereas for m = 2; we obtain an exponential decay rate. (See Theorem 3.2.1). The
main tool used in our proof is an inequality due to Nakao [37]; in which this inequality has been
introduced in order to study the stability of the wave equation, but it is still works in our problem.
Chapter 4:
In this chapter we will prove that if the initial energy E(0) of our solution is negative ( this means
that our initial data are large enough), then our local solutions in bounded and
kutk2 + kruk2 !1
as t tends to +1: In fact it will be proved that the Lp�norm of the solution grows as an exponentialfunction. An essential tool of the proof is an idea used by Gerbi and Said-Houari [15], which based
on an auxiliary function ( which is a small perturbation of the total energy), in order to obtain a
di¤erential inequality leads to the exponential growth result provided that the following conditions
1Z0
g(s)ds <p� 2p� 1 ;
holds.
Chapter 1
Preliminary
Abstract
In this chapter we shall introduce and state some necessary materials needed in the proof of our
results, and shortly the basic results which concerning the Banach spaces, the weak and weak star
topologies, the Lp space, Sobolev spaces and other theorems. The knowledge of all this notations
and results are important for our study.
1
Chapter 1. Preliminary 2
1.1 Banach Spaces - De�nition and Properties
We �rst review some basic facts from calculus in the most important class of linear spaces "Banach
spaces".
De�nition 1.1.1 A Banach space is a complete normed linear space X: Its dual space X 0 is the
linear space of all continuous linear functional f : X ! R:
Proposition 1.1.1 ([43])
X 0 equipped with the norm k:kX0 de�ned by
kfkX0 = sup fjf(u)j : kuk � 1g ; (1.1)
is also a Banach space.
We shall denote the value of f 2 X 0 at u 2 X by either f(u) or hf; uiX0; X :
Remark 1.1.1 ([43]) From X 0 we construct the bidual or second dual X 00 = (X 0)0: Furthermore,
with each u 2 X we can de�ne '(u) 2 X 00 by '(u)(f) = f(u); f 2 X 0, this satis�es clearly
k'(x)k � kuk : Moreover, for each u 2 X there is an f 2 X 0 with f(u) = kuk and kfk = 1, so itfollows that k'(x)k = kuk :
De�nition 1.1.2 Since ' is linear we see that
' : X ! X 00;
is a linear isometry of X onto a closed subspace of X 00, we denote this by
X ,! X 00:
De�nition 1.1.3 If ' ( in the above de�nition) is onto X 00 we say X is re�exive, X u X 00:
Theorem 1.1.1 ([4]; Theorem III.16)
Let X be Banach space. Then, X is re�exive, if and only if,
BX = fx 2 X : kxk � 1g ;
is compact with the weak topology � (X;X 0) : (See the next subsection for the de�nition of � (X;X 0))
De�nition 1.1.4 Let X be a Banach space, and let (un)n2N be a sequence in X. Then un converges
strongly to u in X if and only if
limn!1
kun � ukX = 0;
and this is denoted by un ! u; or limn!1
un = u:
Section 1.1. Banach Spaces - De�nition and Properties
Chapter 1. Preliminary 3
1.1.1 The weak and weak star topologies
Let X be a Banach space and f 2 X 0: Denote by
'f : X ! Rx 7! 'f (x);
(1.2)
when f cover X 0, we obtain a family�'f�f2X0 of applications to X in R:
De�nition 1.1.5 The weak topology on X; denoted by � (X;X 0) ; is the weakest topology on X for
which every�'f�f2X0 is continuous.
We will de�ne the third topology on X 0, the weak star topology, denoted by � (X 0; X) : For all
x 2 X: Denote by'x : X 0 ! R
f 7! 'x(f) = hf; xiX0; X ;(1.3)
when x cover X, we obtain a family ('x)x2X0 of applications to X 0 in R:
De�nition 1.1.6 The weak star topology on X 0 is the weakest topology on X 0 for which every
('x)x2X0 is continuous.
Remark 1.1.2 ([4]) Since X � X 00; it is clear that, the weak star topology � (X 0; X) is weakest
then the topology � (X 0; X 00), and this later is weakest then the strong topology.
De�nition 1.1.7 A sequence (un) in X is weakly convergent to x if and only if
limn!1
f(un) = f(u);
for every f 2 X 0; and this is denoted by un * u.
Remark 1.1.3 ([42]; Remark 1.1.1)
1. If the weak limit exist, it is unique.
2. If un ! u 2 X (strongly), then un * u (weakly):
3. If dimX < +1; then the weak convergent implies the strong convergent.
Proposition 1.1.2 ([43])
On the compactness in the three topologies in the Banach space X :
1- First, the unit ball
B � fx 2 X : kxk � 1g ; (1.4)
Section 1.1. Banach Spaces - De�nition and Properties
Chapter 1. Preliminary 4
in X is compact if and only if dim(X) <1:
2- Second, the unit ball B0 in X 0 (The closed subspace of a product of compact spaces) is weakly
compact in X 0 if and only if X is re�exive.
3- Third, B0 is always weakly star compact in the weak star topology of X 0.
Proposition 1.1.3 ([4]; proposition III.12)
Let (fn) be a sequence in X 0. We have:
1.hfn
�* f in � (X 0; X)
i, [fn(x)! f(x); 8x 2 X] :
2. If fn ! f (strongly), then fn * f , in � (X 0; X 00),
If fn * f in � (X 0; X 00), then fn�* f , in � (X 0; X) :
3. If fn�* f , in � (X 0; X), then kfnk is bounded and kfk � lim inf kfnk :
4. If fn�* f , in � (X 0; X) and xn ! x (strongly) in X, then fn(xn)! f(x):
1.1.2 Hilbert spaces
The proper setting for the rigorous theory of partial di¤erential equation turns out to be the most
important function space in modern physics and modern analysis, known as Hilbert spaces. Then,
we must give some important results on these spaces here.
De�nition 1.1.8 A Hilbert space H is a vectorial space supplied with inner product hu; vi such thatkuk =
phu; ui is the norm which let H complete.
Theorem 1.1.2 ([42], Theorem 1.1.1)
Let (un)n2N is a bounded sequence in the Hilbert space H, then it possess a subsequence which
converges in the weak topology of H:
Theorem 1.1.3 ([42], Theorem 1.1.2)
In the Hilbert space, all sequence which converges in the weak topology is bounded.
Theorem 1.1.4 ([42], Corollary 1.1.1)
Let (un)n2N be a sequence which converges to u; in the weak topology and (vn)n2N is an other sequence
which converge weakly to v; then
limn!1
hvn; uni = hv; ui: (1.5)
Theorem 1.1.5 ([42], Theorem 1.1.3)
Let X be a normed space, then the unit ball
B0 =nl 2 X 0
: klk � 1o; (1.6)
of X0is compact in � (X 0; X).
Section 1.1. Banach Spaces - De�nition and Properties
Chapter 1. Preliminary 5
Proposition 1.1.4 ([42]; Proposition 1.1.1)
Let X and Y be two Hilbert spaces, let (un)n2N 2 X be a sequence which converges weakly to u 2 X;let A 2 L(X;Y ): Then, the sequence (A (un))n2N converges to A(u) in the weak topology of Y:
Proof. For all u 2 X, the functionu 7! hA(u); vi
is linear and continuous, because
jhA(u); vij � kAkL(X; Y ) kukX kvkY ; 8u 2 X; v 2 Y:
So, according to Riesz theorem, there exists w 2 X such that
hA(u); vi = hu;wi, 8u 2 X:
Then,
limn!1
hA (un) ; vi = limn!1
hun; wi
= hu;wi = hA(u); vi: (1.7)
This completes the proof.
Section 1.1. Banach Spaces - De�nition and Properties
Chapter 1. Preliminary 6
1.2 Functional Spaces
1.2.1 The Lp () spaces
De�nition 1.2.1 Let 1 � p � 1; and let be an open domain in Rn; n 2 N: De�ne the standardLebesgue space Lp(); by
Lp() =
8<:f : ! R : f is measurable andZ
jf(x)jp dx <1
9=; : (1.8)
Notation 1.2.1 For p 2 R and 1 � p <1, denote by
kfkp =
0@Z
jf(x)jp dx
1A1p
: (1.9)
If p =1; we have
L1() = ff : ! R : f is measurable and there exists a constant C
such that, jf(x)j � C a.e in g:(1.10)
Also, we denote by
kfk1 = Inf fC; jf(x)j � C a.e in g : (1.11)
Notation 1.2.2 Let 1 � p � 1; we denote by q the conjugate of p i.e.1
p+1
q= 1:
Theorem 1.2.1 ([48])
It is well known that Lp() supplied with the norm k:kp is a Banach space, for all 1 � p � 1:
Remark 1.2.1 In particularly, when p = 2, L2 () equipped with the inner product
hf; giL2() =Z
f(x)g(x)dx; (1.12)
is a Hilbert space.
Theorem 1.2.2 ([43], Corollary 3.2)
For 1 < p <1; Lp() is re�exive space.
Section 1.2. Functional Spaces
Chapter 1. Preliminary 7
Some integral inequalities
We will give here some important integral inequalities. These inequalities play an important role
in applied mathematics and also, it is very useful in our next chapters.
Theorem 1.2.3 ([48], Hölder�s inequality )
Let 1 � p � 1. Assume that f 2 Lp() and g 2 Lq(); then, fg 2 L1() andZ
jfgj dx � kfkp kgkq : (1.13)
Corollary 1.2.1 (Hölder�s inequality - general form)
Lemma 1.2.1 Let f1; f2; :::fk be k functions such that, fi 2 Lpi(); 1 � i � k, and
1
p=1
p1+1
p2+ :::+
1
pk� 1:
Then, the product f1f2:::fk 2 Lp() and kf1f2:::fkkp � kf1kp1 ::: kfkkpk :
Lemma 1.2.2 ( [48], Young�s inequality)
Let f 2 Lp(R) and g 2 Lq(R) with 1 < p < 1; 1 < q < 1 and1
r=1
p+1
q� 1 � 0. Then
f � g 2 Lr(R) andkf � gkLr(R) � kfkLp(R) kgkLq(R) : (1.14)
Lemma 1.2.3 ([43]; Minkowski inequality)
For 1 � p � 1; we have
ku+ vkLp � kukLp + kvkLp : (1.15)
Lemma 1.2.4 ([43])
Let 1 � p � r � q;1
r=�
p+1� �
q; and 1 � � � 1: Then
kukLr � kuk�Lp kuk
1��Lq : (1.16)
Lemma 1.2.5 ([43])
If � () <1; 1 � p � q � 1; then Lq ,! Lp; and
kukLp � �()1p�1q kukLq :
Section 1.2. Functional Spaces
Chapter 1. Preliminary 8
1.2.2 The Sobolev space Wm;p()
Proposition 1.2.1 ([26])
Let be an open domain in RN ; Then the distribution T 2 D0() is in Lp() if there exists a
function f 2 Lp() such that
hT; 'i =Z
f(x)'(x)dx; for all ' 2 D();
where 1 � p � 1; and it�s well-known that f is unique.
De�nition 1.2.2 Let m 2 N and p 2 [0;1] : The Wm;p() is the space of all f 2 Lp(), de�ned
asWm;p() = ff 2 Lp(); such that @�f 2 Lp() for all � 2 Nm such that
j�j =Pn
j=1 �j � m; where, @� = @�11 @�22 ::@
�nn g:
(1.17)
Theorem 1.2.4 ([9])
Wm;p() is a Banach space with their usual norm
kfkWm;p() =Xj�j�m
k@�fkLp ; 1 � p <1, for all f 2 Wm;p(): (1.18)
De�nition 1.2.3 Denote by Wm;p0 () the closure of D() in Wm;p():
De�nition 1.2.4 When p = 2, we prefer to denote by Wm;2() = Hm () and Wm;20 () = Hm
0 ()
supplied with the norm
kfkHm() =
0@Xj�j�m
(k@�fkL2)2
1A 12
; (1.19)
which do at Hm () a real Hilbert space with their usual scalar product
hu; viHm() =Xj�j�m
Z
@�u@�vdx (1.20)
Theorem 1.2.5 ([42]; Proposition 1.2.1)
1) Hm () supplied with inner product h:; :iHm() is a Hilbert space.
2) If m � m0, Hm () ,! Hm0(), with continuous imbedding :
Lemma 1.2.6 ([26])
Since D() is dense in Hm0 () ; we identify a dual H
�m () of Hm0 () in a weak subspace on ;
and we have
D() ,! Hm0 () ,! L2 () ,! H�m () ,! D0();
Section 1.2. Functional Spaces
Chapter 1. Preliminary 9
Lemma 1.2.7 (Sobolev-Poincaré�s inequality)
If
2 � q � 2n
n� 2 ; n � 3
q � 2; n = 1; 2;
then
kukq � C(q; ) kruk2 ; (1.21)
for all u 2 H10 () :
The next results are fundamental in the study of partial di¤erential equations
Theorem 1.2.6 ([9] Theorem 1.3.1)
Assume that is an open domain in RN (N � 1), with smooth boundary �. Then,(i) if 1 � p � n; we have W 1;p � Lq(); for every q 2 [p; p�] ; where p� = np
n� p:
(ii) if p = n we have W 1;p � Lq(); for every q 2 [p; 1) :(iii) if p > n we have W 1;p � L1() \ C0;�(); where � = p� n
p:
Theorem 1.2.7 ([9] Theorem 1.3.2)
If is a bounded, the embedding (ii) and (iii) of theorem 1.1.4 are compacts. The embedding (i) is
compact for all q 2 [p; p�) :
Remark 1.2.2 ([26])
For all ' 2 H2(); �' 2 L2() and for � su¢ ciently smooth, we have
k'(t)kH2() � C k�'(t)kL2() : (1.22)
Proposition 1.2.2 ([43], Green�s formula)
For all u 2 H2(); v 2 H1() we have
�Z
�uvdx =
Z
rurvdx�Z@
@u
@�vd�; (1.23)
where@u
@�is a normal derivation of u at �.
Section 1.2. Functional Spaces
Chapter 1. Preliminary 10
1.2.3 The Lp (0; T;X) spaces
De�nition 1.2.5 Let X be a Banach space, denote by Lp(0; T;X) the space of measurable functions
f : ]0; T [! X
t 7�! f(t)
such that 0@ TZ0
kf(t)kpX dt
1A1p
= kfkLp(0;T;X) <1; for 1 � p <1: (1.24)
If p =1,kfkL1(0;T;X) = sup
t2]0;T [ess kf(t)kX : (1.25)
Theorem 1.2.8 ([42])
The space Lp(0; T;X) is complete.
We denote by D0 (0; T;X) the space of distributions in ]0; T [ which take its values in X, and let usde�ne
D0 (0; T;X) = L (D ]0; T [ ; X) ;
where L (�; ') is the space of the linear continuous applications of � to ': Since u 2 D0 (0; T; X) ;we de�ne the distribution derivation as
@u
@t(') = �u
�d'
dt
�; 8' 2 D (]0; T [) ; (1.26)
and since u 2 Lp (0; T; X) ; we have
u(') =
TZ0
u(t)'(t)dt; 8' 2 D (]0; T [) : (1.27)
We will introduce some basic results on the Lp(0; T; X) space. These results, will be very useful in
the other chapters of this thesis.
Lemma 1.2.8 ([26] Lemma 1.2 )
Let f 2 Lp(0; T; X) and@f
@t2 Lp(0; T; X); (1 � p � 1) ; then, the function f is continuous from
[0; T ] to X:i.e: f 2 C1(0; T; X):
Section 1.2. Functional Spaces
Chapter 1. Preliminary 11
Lemma 1.2.9 ([26])
Let ' = ]0; T [� an open bounded domain in R�Rn; and let g�; g are two functions in Lq (]0; T [ ; Lq()) ;1 < q <1 such that
kg�kLq(0;T;Lq()) � C; 8� 2 N (1.28)
and
g� ! g in ';
then
g� * g in Lq (') :
Theorem 1.2.9 ([9]; Proposition 1.4.17)
Lp(0; T; X) equipped with the norm k:kLp(0;T;X), 1 � p � 1 is a Banach space.
Proposition 1.2.3 ([14])
Let X be a re�exive Banach space, X 0 it�s dual, and 1 � p <1; 1 � q <1;1
p+1
q= 1: Then the
dual of Lp(0; T; X) is identify algebraically and topologically with Lq(0; T;X 0):
Proposition 1.2.4 ([9])
Let X; Y be to Banach space, X � Y with continuous embedding, then we have Lp(0; T; X) � Lp(0;
T; Y ) with continuous embedding.
The following compactness criterion will be useful for nonlinear evolution problems, especially in
the limit of the non linear terms.
Proposition 1.2.5 ([26]).
Let B0; B; B1 be Banach spaces with B0 � B � B1; assume that the embedding B0 ,! B is compact
and B ,! B1 are continuous. Let 1 < p < 1; 1 < q < 1; assume further that B0 and B1 are
re�exive.
De�ne
W � fu 2 Lp (0; T; B0) : u0 2 Lq (0; T; B1)g : (1.29)
Then, the embedding W ,! Lp (0; T; B) is compact.
Section 1.2. Functional Spaces
Chapter 1. Preliminary 12
1.2.4 Some Algebraic inequalities
Since our study based on some known algebraic inequalities, we want to recall few of them here.
Lemma 1.2.10 ([48];The Cauchy-Schwarz inequality)
Every inner product satis�es the Cauchy-Schwarz inequality
hx1 ; x2i � kx1k kx2k : (1.30)
The equality sign holds if and only if x1 and x2 are dependent.
Young�s inequalities :
Lemma 1.2.11 For all a; b 2 R+, we have
ab � �a2 +b2
4�; (1.31)
where � is any positive constant.
Proof. Taking the well-known result
(2�a� b)2 � 0 8a; b 2 R
for all � > 0, we have
4�2a2 + b2 � 4�ab � 0:
This implies
4�ab � 4�2a2 + b2
consequently,
ab � �a2 +1
4�b2:
This completes the proof.
Lemma 1.2.12 ([43])
For all a; b � 0; the following inequality holds
ab � ap
p+bq
q;
where,1
p+1
q= 1:
Section 1.2. Functional Spaces
Chapter 1. Preliminary 13
Proof. Let G = (0; 1), and f : G! R is integrable, such that
f(x) =
8><>:p log a, 0 � x � 1
p
q log b,1
p� x � 1
;
for all a; b � 0 and 1p+1
q= 1:
Since '(t) = et is convex, and using Jensen�s inequality
'
0@ 1
� (G)
ZG
f(x)dx
1A � 1
� (G)
ZG
' (f(x)) dx: (1.32)
Consequently, we have
1
� (G)
ZG
' (f(x)) dx =
1Z0
ef(x)dx =
1=pZ0
ep log adx+
1Z1=p
eq log bdx
=
1=pZ0
apdx+
1Z1=p
bqdx
=1
p(ap) +
�1� 1
p
�bq:
(1.33)
=ap
p+bq
q;
where, � (G) = 1 and
'
0@ 1
� (G)
ZG
f(x)dx
1A = e(R 10 f(x)dx) = e(
R 1=p0 p log adx+
R 11=p q log bdx)
= e(log a+log b) = elog ab
= ab: (1.34)
Using (1:32), (1:33) and (1:34) to conclude the result.
Section 1.2. Functional Spaces
Chapter 1. Preliminary 14
1.3 Existence Methods
1.3.1 The Contraction Mapping Theorem
Here we prove a very useful �xed point theorem called the contraction mapping theorem. We will
apply this theorem to prove the existence and uniqueness of solutions of our nonlinear problem.
De�nition 1.3.1 Let f : X ! X be a map of a metric space to itself. A point x 2 X is called a
�xed point of f if f(x) = x.
De�nition 1.3.2 Let (X; dX) and (Y; dY ) be metric spaces. A map ' : X ! Y is called a contrac-
tion if there exists a positive number C < 1 such that
dY ('(x); '(y)) � CdX(x; y); (1.35)
for all x; y 2 X.
Theorem 1.3.1 (Contraction mapping theorem [45] )
Let (X; d) be a complete metric space. If ' : X ! X is a contraction, then ' has a unique �xed
point.
1.3.2 Gronwell�s lemma
Theorem 1.3.2 ( In integral form)
Let T > 0; and let ' be a function such that, ' 2 L1(0; T ); ' � 0; almost everywhere and � be
a function such that, � 2 L1(0; T ); � � 0; almost everywhere and �' 2 L1 [0; T ] ; C1; C2 � 0.
Suppose that
�(t) � C1 + C2
tZ0
'(s)�(s)ds; for a.e t 2 ]0; T [ ; (1.36)
then,
�(t) � C1 exp
0@C2 tZ0
'(s)ds
1A ; for a.e t 2 ]0; T [ : (1.37)
Proof. Let
F (t) = C1 + C2
tZ0
'(s)�(s)ds; for t 2 [0; T ] ; (1.38)
we have,
�(t) � F (t);
Section 1.3. Existence Methods
Chapter 1. Preliminary 15
From (1:38) we have
F 0(t) = C2'(t)�(t)
� C2'(t)�(t); for a.e t 2 ]0; T [ : (1.39)
Consequently,
d
dt
8<:F (t) exp0@� tZ
0
C2'(s)ds
1A9=; � 0; (1.40)
then,
F (t) � C1 exp
0@C2 tZ0
'(s)ds
1A ; for a.e t 2 ]0; T [ : (1.41)
Since � � F; then our result holds.
In particle, if C1 = 0; we have � = 0 for almost everywhere t 2 ]0; T [ :
1.3.3 The mean value theorem
Theorem 1.3.3 Let G : [a; b]! R be a continues function and ' : [a; b]! R is an integral positivefunction, then there exists a number x in (a; b) such that
bZa
G(t)'(t)dt = G(x)
bZa
'(t)dt: (1.42)
In particular for '(t) = 1; there exists x 2 (a; b) such that
bZa
G(t)dt = G(x) (b� a) : (1.43)
Proof. Let
m = inf fG(x); x 2 [a; b]g (1.44)
and
M = sup fG(x); x 2 [a; b]g (1.45)
of course m and M exist since [a; b] is compact.
Then, it follows that
m
bZa
'(t)dt �bZa
G(t)'(t)dt �M
bZa
'(t)dt: (1.46)
Section 1.3. Existence Methods
Chapter 1. Preliminary 16
By monotonicity of the integral. dividing through byR ba'(t)dt; we have that
m �R baG(t)'(t)dtR ba'(t)dt
�M: (1.47)
Since G(t) is continues, the intermediate value theorem implies that there exists x 2 [a; b] such that
G(x) =
R baG(t)'(t)dtR ba'(t)dt
: (1.48)
Which completes the proof.
Section 1.3. Existence Methods
Chapter 2
Local Existence
Abstract
Our goal in this chapter is to study the local existence ( local well-possedness) of the problem (P );
for u in C ([0; T ] ; H10 ()). For this purpose we consider, �rst the related problem for u �xed in
C ([0; T ] ; H10 ()) 8>>>>>>>>>>>><>>>>>>>>>>>>:
vtt ��vt ��v +R t0g(t� s)�v(s; x)ds
+ jvtjm�2 vt = jujp�2 u; x 2 ; t > 0
v (0; x) = u0 (x) ; vt (0; x) = u1 (x) ; x 2
v (t; x) = 0; x 2 �; t > 0
; (2.1)
and we will prove the local existence of this problem by using the Faedo-Galerkin method. Then,
by using the well-known contraction mapping theorem, we can show the local existence of (P ):
Our techniques of proof follows carefully the techniques due to Georgiev and Todorova [14]; with
necessary modi�cations imposed by the nature of our problem. The �rst step of our proof is the
choice of the space where the local solution exists. The minimal requirement for this space is that
u(t; x) be time continuous. The weak space satisfying the above requirement is C ([0; T ] ; H), where
H = H10 ()�H1
0 () is the natural energy space for (P ).
17
Chapter 2. Local Existence 18
2.1 Local Existence Result
In order to prove our local existence results, let us introduce the following space
YT =
8>><>>:u : u 2 C ([0; T ] ; H1
0 ()) ;
ut 2 C([0; T ] ; H10 ()) \ Lm([0; T ]� )
9>>=>>; : (2.2)
Our main result in this chapter reads as follows:
Theorem 2.1.1 Let (u0; u1) 2 (H10 ())
2 be given. Suppose that m � 2; p � 2 be such that
max fm; pg � 2 (n� 1)n� 2 ; n � 3: (2.3)
Then, under the conditions (G1) and (G2), the problem (P ) has a unique local solution u(t; x) 2 YT ,for T small enough.
The proof of theorem 2.1.1 will be established through several lemmas. The presence of the term
jujP�2 u in the right hand side of our problem (P ) ; gives us negative values of the energy. For this
purpose we �xed u 2 C ([0; T ] ; H10 ()) in the right hand side of (P ) and we will prove that our
problem (2:1) ; admits a solution.
Lemma 2.1.1 ([14]; Theorem 2.1) Let (u0; u1) 2 (H10 ())
2, assume that m � 2; p � 2 and (2:3)holds. Then, under the conditions (G1) and (G2), there exists a unique weak solution v 2 YT to theproblem (2:1); for any u 2 C ([0; T ] ; H1
0 ()) given.
The proof of the above Lemma follows the techniques due to Lions [26], in order to deal with the
convergence of the non linear terms in our problem, we must take �rst our initial data (u0; u1) in a
high regularity (that is, u0 2 H2() \H10 (); and u1 2 H1
0 () \ L2(m�1)()).
Lemma 2.1.2 ([26]; Theorem 3.1) Let u 2 C ([0; T ] ; H10 ()). Suppose that
u0 2 H2() \H10 (); (2.4)
u1 2 H10 () \ L2(m�1)(); (2.5)
assume further that m � 2; p � 2. Then, under the conditions (G1) and (G2) there exists a uniquesolution v of the problem (2:1) such that
v 2 L1�[0; T ] ; H2() \H1
0 ()�; (2.6)
Section 2.1. Local Existence Result
Chapter 2. Local Existence 19
vt 2 L1�[0; T ] ; H1
0 ()�; (2.7)
vtt 2 L1�[0; T ] ; L2()
�; (2.8)
vt 2 Lm ([0; T ]� ()) : (2.9)
The following technical Lemma will play an important role in the sequel.
Lemma 2.1.3 For any v 2 C1 (0; T;H2()) we have
Z
tZ0
g(t� s)�v(s):v0(t)dsdx =1
2
d
dt(g � rv) (t)� 1
2
d
dt
24 tZ0
g(s)
Z
jrv(t)j2 dxds
35�12(g0 � rv) (t) + 1
2g(t)
Z
jrv(t)j2 dxds:
where (g � u)(t) =tZ0
g(t� s)
Z
ju(s)� u(t)j2 dxds:
The proof of this result is given in [31], for the reader�s convenience we repeat the steps here.
Proof. It�s not hard to see
Z
tZ0
g(t� s)�v(s):v0(t)dsdx = �tZ0
g(t� s)
Z
rv0(t):rv(s)dxds
= �tZ0
g(t� s)
Z
rv0(t): [rv(s)�rv(t)] dxds
�tZ0
g(t� s)
Z
rv0(t):rv(t)dxds:
Consequently,
Z
tZ0
g(t� s)�v(s):v0(t)dsdx =1
2
tZ0
g(t� s)d
dt
Z
jrv(s)�rv(t)j2 dxds
�tZ0
g(s)
0@ d
dt
1
2
Z
jrv(t)j2 dx
1A ds
Section 2.1. Local Existence Result
Chapter 2. Local Existence 20
which implies,Z
tZ0
g(t� s)�v(s):v0(t)dsdx =1
2
d
dt
24 tZ0
g(t� s)
Z
jrv(s)�rv(t)j2 dxds
35�12
d
dt
24 tZ0
g(s)
Z
jrv(t)j2 dxds
35�12
tZ0
g0(t� s)
Z
jrv(s)�rv(t)j2 dxds
+1
2g(t)
Z
jrv(t)j2 dxds:
This completes the proof.
Proof of Lemma 2.1.2.
Existence:
Our main tool is the Faedo-Galerkin�s method, which consist to construct approximations of the so-
lutions, then we obtain a prior estimates necessary to guarantee the convergence of approximations.
Our proof is organized as follows. In the �rst step, we de�ne an approach problem in bounded
dimension space Vn which having unique solution vn and in the second step we derive the various
a priori estimates. In the third step we will pass to the limit of the approximations by using the
compactness of some embedding in the Sobolev spaces.
1. Approach solution:
Let V = H10 ()\H2() the separable Hilbert space. Then there exists a family of subspaces fVng
such that
i) Vn � V (dimVn <1); 8n 2 N:ii) Vn ! V; such that, there exist a dense subspace # in V and for all v 2 #; we can get sequencefvngn2N 2 Vn; and vn ! v in V:
iii) Vn � Vn+1 and [n2N�Vn = H10 () \H2():
For every n � 1; let Vn = Span fw1; :::; wng ; where fwig, 1 � i � n; is the orthogonal complete
system of eigenfunctions of �� such that kwjk2 = 1, wj 2 H2() \ L2(m�1)() for all j = 1; :::; n.Denote by f�jg the related eigenvalues, where wj are solutions of the following initial boundaryvalue problem 8>><>>:
��wj = �jwj
wj = 0 on �
j = 1; :::; in : (2.10)
Section 2.1. Local Existence Result
Chapter 2. Local Existence 21
According to (iii) ; we can choose vn0; vn1 2 [w1; :::; wn] such that
vn0 �nXj=1
�jnwj �! u0 in H10 () \H2(), (2.11)
vn1 �nXj=1
�jnwj �! u1 in H10 () \ L2(m�1)(); (2.12)
where�jn =
Ru0wjdx
�jn =Ru1wjdx:
We seek n functions 'n1 ; :::; 'nn 2 C2 [0; T ] ; such that
vn(t) =nXj=1
'nj (t)wj(x); (2.13)
solves the problem 8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
R
((v00n(t)��v0n(t)��vn(t)) �dx
+R
�tR0
g(t� s)�vn(s)ds+ jv0n(t)jm�2 v0n(t)
��dx
=R
ju(t)jp�2 u(t)�dx
vn (0) = vn0; v0n (0) = vn1
; (2.14)
where the prime "0" denotes the derivative with respect to t. For every � 2 Vn and t � 0: Taking� = wj; in (2:14) yields the following Cauchy problem for a ordinary di¤erential equation with
unknown 'nj : 8>>>>>>>>><>>>>>>>>>:
'00nj (t) + �j'0nj (t) + �j'
nj (t) + �j
tR0
g(t� s)'nj (s)ds
+��'0hj (t)��m�2 '0nj (t) = j(t)
'nj (0) =R
u0wj; '0nj (0) =
R
u1wj; j = 1; :::; n
; (2.15)
for all j; where
j(t) =
Z
ju(t)jp�2 u(t)wj 2 C [0; T ] :
Section 2.1. Local Existence Result
Chapter 2. Local Existence 22
By using the Caratheodory theorem for an ordinary di¤erential equation, we deduce that, the
above Cauchy problem yields a unique global solution 'nj 2 H3 [0; T ] ; and by using the embedding
Hm [0; T ] ,! Cm�1 [0; T ], we deduce that the solution 'nj 2 C2 [0; T ]. In turn, this gives a unique
vn de�ned by (2:13) and satisfying (2:14):
2. The a priori estimates:
The next estimate prove that the energy of the problem (2:1) is bounded and by using a result in
[9]; we conclude that; the maximal time tn of existence of (2:15) can be extended to T .
The �rst a priori estimate:
Substituting � = v0n(t) into (2:14), we obtainZ
v00n(t)v0n(t)dx�
Z
�v0n(t)v0n(t)dx�
Z
�vn(t)v0n(t)dx
+
Z
tZ0
g(t� s)�vn(s)dsv0n(t)dx+
Z
jv0n(t)jm�2
v0n(t)v0n(t)dx (2.16)
=
Z
f(u)v0n(t)dx:
for every n � 1; where f(u) = ju(t)jp�2 u(t); Since the following mapping
L2() �! L2()
u 7! jujp�2 u
is continues, we deduce that,
jujp�2 u 2 L1�[0; T ] ; L2()
�:
So, f 2 H1 ([0; T ] ; H1()). Consequently, by using the Lemma 2.1.3 and (G1) we get easily
1
2
d
dtkv0n(t)k
22 + krv0n(t)k
22 +
1
2
d
dtkrvn(t)k22
+1
2
d
dt
24 tZ0
g(t� s)
Z
jrvn(s)�rvn(t)j2 dxds
35� 12
d
dt
0@krvn(t)k22 tZ0
g(s):ds
1A�12
tZ0
g0(t� s)
Z
jrvn(s)�rvn(t)j2 dxds+1
2g(t) krvn(t)k22 + kv0n(t)k
mm
=
Z
f(u):v0n(t)dx:
Section 2.1. Local Existence Result
Chapter 2. Local Existence 23
Therefore, we obtain
d
dtEn(t)�
Z
f(u):v0n(t)dx+ krv0n(t)k22 + kv0n(t)k
mm
=1
2(g0 � rvn)(t)�
1
2g(t) krvn(t)k22 ;
where
En(t) =1
2
8<:kv0n(t)k22 +0@1� tZ
0
g(s)ds
1A krvn(t)k22 + (g � rvn)(t)9=; ;
is the functional energy associated to the problem (2:1):
It�s clear by using (G2), that
d
dtEn(t)�
Z
f(u):v0n(t)dx+ krv0n(t)k22 + kv0n(t)k
mm � 0; 8t � 0:
Which implies, by using Young�s inequality, for all � > 0
d
dtEn(t) + krv0n(t)k
22 + kv0n(t)k
mm �
1
4�kf(u)k22 + � kv0n(t)k
22 : (2.17)
Integrating (2:17) over [0; t] ; (t < T ), we obtain
En(t) +
tZ0
krv0n(s)k22 ds+
tZ0
kv0n(s)kmm ds
� 1
4�
TZ0
kf(u)k22 ds+ �
TZ0
kv0n(s)k22 ds+
1
2
�ku1nk22 + kru0nk
22
�:
Since f 2 H1(0; T;H10 ()); we deduce
En(t) +
tZ0
krv0n(s)k22 ds+
tZ0
kv0n(s)kmm ds � CT
�ku1nk22 + kru0nk
22
�; (2.18)
for, � small enough and every n � 1, where CT > 0 is positive constant independent of n:Then, by the de�nition of En(t) and by using (2:11) ; (2:12), we get
kv0n(t)k22 +
0@1� tZ0
g(s)ds
1A krvn(t)k22 + (g � rvn)(t) � KT ; (2.19)
andtZ0
kv0n(s)kmm ds � KT ; (2.20)
Section 2.1. Local Existence Result
Chapter 2. Local Existence 24
and
tZ0
krv0n(s)k22 ds � KT ; (2.21)
where KT = CT�ku1nk22 + kru0nk
22
�; by (2:19), we get tn = T; 8n:
However, the insu¢ cient regularity of the nonlinear operator, jvtjm�2 vt; with the presence of theviscoelastic term and strong damping, we must prove in the next a prior estimates that, the family of
approximations vn de�ned in (2:13) is compact in the strong topology and by using compactness of
the embedding H1([0; T ] ; H1()) ,! L2([0; T ] ; L2()); we can extract a subsequence of vn denoted
also by v� such that v0� converges strongly in L
2([0; T ] ; L2()): To do this, it�s su¢ ces to prove
that v0n is bounded in L
1([0; T ] ; H10 ()) and v
00
n is bounded in L1([0; T ] ; L2()), then by using
Aubin-Lions Lemma, our conclusion holds.
The second a priori estimate:
Substituting � = wj in (2:14) and taking ��wj = �jwj, multiplying by '0nj (t) and summing up the
product result with respect to j; we get by Green�s formula.
Z
rv00n:rv0ndx+Z
�v0n:�v0ndx+
Z
�vn:�v0ndx�
Z
tZ0
g(t� s)�vn:�v0ndsdx
+nXi=1
Z
@
@xi
�jv0nj
m�2v0n
� @v0n@xi
dx (2.22)
=
Z
rf(u):rv0ndx:
As in [26], we have
nXi=1
Z
@
@xi
�jv0nj
m�2v0n
� @v0n@xi
dx
= (m� 1)nXi=1
Z
@
@xi
�jv0nj
m�22
@v0n@xi
�2dx (2.23)
=4(m� 1)
m2
nXi=1
Z
�@
@xi
�jv0nj
m�22 v0n
��2dx:
Section 2.1. Local Existence Result
Chapter 2. Local Existence 25
Also, the fourth term in the left hand side of (2:22) can be written as followsZ
tZ0
g(t� s)�vn:�v0ndsdx = �1
2g(t) k�vnk22 +
1
2(g0 ��vn)
�12
d
dt
8<:(g ��vn)� k�vnk22tZ0
g(s)ds
9=; : (2.24)
Therefore, (2:22) becomes
1
2
d
dt
24krv0nk22 + (g ��vn) + k�vnk220@1� tZ
0
g(s)ds
1A35+4(m� 1)
m2
nXi=1
Z
�@
@xi
�jv0nj
m�22 v0n
��2dx (2.25)
+ k�v0nk22 +
1
2g(t) k�vnk22 �
1
2(g0 ��vn)
=
Z
rf(u):rv0ndx:
Let us de�ne the energy term
Kn(t) =1
2
24krv0nk22 + (g ��vn) + k�vnk220@1� tZ
0
g(s)ds
1A35 (2.26)
Then, it�s clear that (2:25) takes the form
d
dtKn(t)�
Z
rf(u):rv0ndx+ k�v0nk22
+nXi=1
Z
�@
@xi
�jv0nj
m�22 v0n
��2dx
(2.27)
= �12(g(t) k�vnk22) +
1
2(g0 ��vn):
Using (G1) ; (G2) and integrating (2:27) over [0; t] ; we obtain
Kn(t) +nXi=1
tZ0
Z
�@
@xi
�jv0nj
m�22 v0n
��2dxds+
tZ0
k�v0nk22 ds
�TZ0
Z
rf(u):rv0ndxds+1
2(krv1nk22 + k�v0nk
22): (2.28)
Section 2.1. Local Existence Result
Chapter 2. Local Existence 26
Obviously, by using Young�s inequality, we get
TZ0
Z
rf(u):rv0ndxds �1
4�
TZ0
krf(u)k22 ds+ �
TZ0
krv0nk22 ds:
Inserting the above estimate into (2:28); to get
Kn(t) +
nXi=1
tZ0
Z
�@
@xi
�jv0nj
m�22 v0n
��2dxds+
tZ0
k�v0nk22 ds
� CT (krv1nk22 + k�v0nk22):
Thus,
Kn(t) +nXi=1
tZ0
Z
�@
@xi
�jv0nj
m�22 v0n
��2dxds+
tZ0
k�v0nk22 ds � CT ;
for, � small enough and every n � 1, where CT > 0 is positive constant independent of n: Therefore,this equivalent by the de�nition of Kn(t)
krv0nk22 + (g ��vn) + k�vnk
22
0@1� tZ0
g(s)ds
1A � CT ; (2.29)
andnXi=1
tZ0
Z
�@
@xi
�jv0nj
m�22 v0n
��2dxds � CT ; (2.30)
andtZ0
k�v0nk22 ds � CT : (2.31)
Then, from (2:29), (2:30) and (2:31) ; we conclude
v0n is bounded in L1([0; T ] ; H1
0 ()); (2.32)
vn is bounded in L1([0; T ] ; H2()); (2.33)
@
@xi
�jv0nj
m�22 v0n
�is bounded in L2([0; T ] ; L2()); i = 1; :::; n: (2.34)
The third a priori estimate:
It�s clear that
d
dt
24 tZ0
g(t� s)�vn(s)ds
35 = g(0)�vn +
tZ0
g0(t� s)�vn(s)ds: (2.35)
Section 2.1. Local Existence Result
Chapter 2. Local Existence 27
Performing an integration by parts in (2:35) we �nd that
d
dt
24 tZ0
g(t� s)�vn(s)ds
35 = g(t)�un0 +
tZ0
g(t� s)�v0n(s)ds: (2.36)
Now, returning to (2:14), di¤erentiating throughout with respect to t, and using (2:36), we obtainZ
(v000n (t)��v00n(t)��v0n(t)) �dx
+
Z
0@ tZ0
g(t� s)�v0n(s)ds+ g(t)�un0 + (m� 1)(jv0n(t)jm�2
v00n(t))
1A �dx
=
Z
(f(u))0)�dx; (2.37)
where, (f(u))0 =@f
@t. By substitution of � = v00h(t) in (2:37); yieldsZ
v000n (t)v00n(t)dx�
Z
�v00n(t)v00n(t)dx�
Z
�v0n(t)v00n(t)dx
+
Z
0@ tZ0
g(t� s)�v0n(s)ds+ g(t)�un0
1A v00n(t)dx
+(m� 1)Z
jv0n(t)jm�2
v00n(t)v00n(t)dx (2.38)
=
0Z
(f(u))0 v00n(t)dx:
The fourth term in the left hand side of (2:38) can be analyzed as follows. It�s clear that Lemma
2.1.3 implies
Z
tZ0
g(t� s)�v0n(s):v00n(t)dsdx = �
Z
tZ0
g(t� s)rv0n(s):rv00n(t)dsdx
=1
2
d
dt(g � rv0n) (t)�
1
2
d
dt
tZ0
g(s) krv0n(t)k22 ds
(2.39)
�12(g0 � rv0n) (t) +
1
2g(t) krv0n(t)k
22 :
Section 2.1. Local Existence Result
Chapter 2. Local Existence 28
Also,
(m� 1)hjv0n(t)jm�2
v00n(t); v00n(t)i =
4(m� 1)m2
Z
�@
@t
�jv0nj
m�22 v0n(t)
��2dx: (2.40)
Inserting (2:39) and (2:40) into (2:38), we obtain
1
2
d
dtkv00nk
22 + krv00nk
22 +
1
2
d
dtkrv0nk
22 +
1
2
d
dt
8<:(g � rv0n) (t)� krv0nk22tZ0
g(s)ds
9=;+4(m� 1)
m2
Z
�@
@t
�jv0nj
m�22 v0n(t)
��2dx+ (g(t)�un0)
Z
v00n(t)dx (2.41)
=
Z
(f(u))0 v00n(t)dx+1
2(g0 � rv0n) (t)�
1
2g(t) krv0nk
22 :
Let us denote by
n(t) =1
2
8<:kv00nk22 ++(g � rv0n) (t) + (1�tZ0
g(s)ds) krv0nk22
9=; : (2.42)
We obtain, from (2:41) ; (G2) that
d
dtn(t)�
Z
(f(u))0 :v00n(t)dx+4(m� 1)
m2
Z
�@
@t
�jv0nj
m�22 v0n(t)
��2dx
+ krv00nk22 + (g(0)�un0)
Z
v00n(t)dx
=1
2(g0 � rv0n) (t)�
1
2g(t) krv0n(t)k
22
� 0:
Integration the above estimate over [0; t], we conclude
n(t) +4(m� 1)
m2
tZ0
Z
�@
@t
�jv0nj
m�22 v0n(t)
��2dxds
+
tZ0
krv00nk22 ds+ (g(0)�un0)
tZ0
Z
v00n(s)dxds
� 1
2kv00n0k
22 +
1
2krvn1k22 +
1
4�
TZ0
(f(u))0 22ds+ �
TZ0
kv00nk22 ds:
Section 2.1. Local Existence Result
Chapter 2. Local Existence 29
Then, for � small enough, we deduce
n(t) +4(m� 1)
m2
tZ0
Z
�@
@t
�jv0nj
m�22 v0n
��2dxds
+
tZ0
krv00nk22 ds+ (g(0)�un0)
tZ0
Z
v00ndxds (2.43)
� CT
�kv00n0k
22 + krvn1k
22
�:
In order to estimate the term kv00n0k22, taking t = 0 in (2:14), we �nd
kv00n0k22 =
Z
�vn1:v00n0dx+
Z
�vn0:v00n0dx
�Z
jvn1jm�2 vn1:v00n0dx+Z
f(u(0)):v00n0dx:
Thanks to Cauchy-Schwartz inequality (Lemma 1.2.10), we write
kv00n0k22 � kv00n0k2
0BB@k�vn1k2 + k�vn0k2 + kf(u(0))k2 +0@Z
jv1nj2(m�1) dx
1A12
1CCA ;
which implies, by using (2:11) and (2:12) ; that
kv00n0k2 � k�vn1k2 + k�vn0k2 + kf(u(0))k2 +
0@Z
jv1nj2(m�1) dx
1A12
� C:
(2.44)
Then,
n(t) � LT ; (2.45)
4(m� 1)m2
tZ0
Z
�@
@t
�jv0nj
m�22 v0n
��2dxds � LT ; (2.46)
andtZ0
krv00nk22 ds � LT ; (2.47)
andtZ0
Z
v00ndxds � LT : (2.48)
Section 2.1. Local Existence Result
Chapter 2. Local Existence 30
where LT > 0:
From (2:11); (2:12) and (2:44)� (2:48) ; we deduce
v00n is bounded in L1([0; T ] ; L2()); (2.49)
vn is bounded in L1([0; T ] ; H10 ()); (2.50)
@
@t
�jv0nj
m�22 v0n
�is bounded in L2([0; T ] ; L2()); i = 1; :::; n: (2.51)
3.Pass to the limit:
By the �rst, the second and the third estimates, we obtain
vn is bounded in L1([0; T ] ; H2() \H10 ()); (2.52)
v0n is bounded in L1([0; T ] ; H1
0 ()); (2.53)
v00n is bounded in L1([0; T ] ; L2()); (2.54)
v0n is bounded in Lm((0; T )� ): (2.55)
Therefore, up to a subsequence, and by using the (Theorem 1.1.5), we observe that there exists a
subsequence v� of vn and a function v that we my pass to the limit in (2:14); we obtain a weak
solution v of (2:1) with the above regularity
v� *� v in L1([0; T ] ; H1
0 () \H2 ());
v0� *� v0in L1([0; T ] ; H1
0 () \ Lm());
v00� *� v
00in L1([0; T ] ; L2()):
By using the fact that
L1([0; T ] ; L2()) ,! L2([0; T ] ; L2());
L1([0; T ] ; H10 ()) ,! L2([0; T ] ; H1
0 ()):
We get
v0n is bounded in L2([0; T ] ; H1
0 ());
v00n is bounded in L2([0; T ] ; L2());
therefore,
v0n is bounded in H1([0; T ] ; H1()): (2.56)
Consequently, since the embedding
H1([0; T ] ; H1()) ,! L2([0; T ] ; L2())
Section 2.1. Local Existence Result
Chapter 2. Local Existence 31
is compact, then we can extract a subsequence v0� such that
v0� ! v0 in L2([0; T ] ; L2()). (2.57)
which implies
v0� ! v0 a.e on (0; T )� .
By (2:20), we have
v0n is bounded in Lm([0; T ]� ):
and by using Theorem 1.2.2
jv0� jm�2
v0� * # in Lm0([0; T ] ; Lm
0())
wherem
m� 1 = m0:
The estimates (2:34) and (2:46) imply that
jv0� jm�22 v0� * � in H1([0; T ] ; H1()
by using the fact that, the mapping u 7�! jujm�2 u is continuous, (Lemma 1.2.9) and since the weaktopology is separate, we deduce
# = jv0jm�2 v0 (2.58)
� = jv0jm�22 v0 (2.59)
Then, by using the uniqueness of limit, we deduce
jv0� jm�2
v0� * jv0jm�2 v0 in Lm0([0; T ] ; Lm
0()) (2.60)
jv0� jm�22 v0� * jv0j
m�22 v0 in H1([0; T ] ; H1() (2.61)
Now, we will pass to the limit in (2:14), by the same techniques as in [26]:
Taking � = wj, n = � and �xed j < �;Z
v00� (t):wjdx+
Z
rv� (t):rwjdx+Z
rv0� (t):rwjdx
�Z
tZ0
g(t� s)rv� (s):rwjdsdx+Z
jv0� (t)jm�2
v0� (t):wjdx (2.62)
=
Z
f(u):wjdx:
Section 2.1. Local Existence Result
Chapter 2. Local Existence 32
We obtain, by using the property of continuous of the operator in the distributions spaceZ
v00� (t):wjdx *�Z
v00(t):wjdx; in D0 (0; T )
Z
rv� (t):rwjdx *�Z
rv(t):rwjdx; in L1 (0; T )Z
rv0� (t):rwjdx *�Z
rv0(t):rwjdx; in L1 (0; T )
Z
tZ0
g(t� s)rv� (s):rwjdsdx *�Z
tZ0
g(t� s)rv(s):rwjdsdx; in L1 (0; T )Z
jv0� (t)jm�2
v0� (t):wjdx *�Z
jv0(t)jm�2 v0(t):wjdx; in L1 (0; T )
We deduce from (2:62), thatZ
v00(t):wjdx+
Z
rv(t):rwjdx+Z
rv0(t):rwjdx
�Z
tZ0
g(t� s)rv(s):rwjdsdx+Z
jv0(t)jm�2 v0(t):wjdx (2.63)
=
Z
f(u):wjdx:
Since, the basis wj (j = 1; :::) is dense in H10 () \H2 () ; we can generalize (2:63) ; as followsZ
v00(t):'dx+
Z
rv(t):r'dx+Z
rv0(t):r'dx
�Z
tZ0
g(t� s)rv(s):r'dsdx+Z
jv0(t)jm�2 v0(t):'dx
=
Z
f(u):'dx; 8' 2 H10 () \H2 () :
Then,
v 2 L1�[0; T ] ; H2() \H1
0 ()�;
vt 2 L1�[0; T ] ; H1
0 ()�;
vtt 2 L1�[0; T ] ; L2()
�;
vt 2 Lm ([0; T ]� ()) :
This complete the our proof of existence.
Section 2.1. Local Existence Result
Chapter 2. Local Existence 33
Uniqueness:
Let v1; v2 two solutions of (2:1), and let w = v1 � v2 satisfying :
w00 ��w ��w0 +tZ0
g(t� s)�wds+�jv01j
m�2v01 � jv02j
m�2v02
�= 0: (2.64)
Multiplying (2:64), by w0and integrating over , we get
1
2
d
dt
0@kw0n(t)k22 +0@1� tZ
0
g(s)ds
1A krwn(t)k22 + (g � rwn)(t)1A
+
Z
�jv01j
m�2v01 � jv02j
m�2v02
�w0dx
= �krw0n(t)k22 +
1
2(g0 � rwn)(t)�
1
2g(t) krwn(t)k22 :
Denote by
J(t) = kw0n(t)k22 +
0@1� tZ0
g(s)ds
1A krwn(t)k22 + (g � rwn)(t): (2.65)
Since the function y 7! jyjm�2 y is increasing, we haveZ
�jv01j
m�2v01 � jv02j
m�2v02
�w0dx � 0
and since1
2(g0 � rwn)(t) � 0;
we deduce �d
dtJ(t) � 0
�: (2.66)
This implies that J(t) is uniformly bounded by J(0) and is decreasing in t; since w(0) = 0; we
obtain w = 0 and v1 = v2:
Proof of Lemma 2.1.1.
As in [14]; since D() = H2(), we approximate, u0; u1 by sequences (u�0) ; (u�1) in D(), and uby a sequence (u�) in C ([0; T ] ;D()), for the problem (2:1). Lemma 2.1.2 guarantees the existenceof a sequence of unique solutions (v�) satisfying (2:6)� (2:9) : Now, to complete the proof of Lemma2.1.1, we proceed to show that the sequence (v�) is Cauchy in YT equipped with the norm
kuk2YT = kuk2H + kutk
2Lm([0;T ]�) :
Section 2.1. Local Existence Result
Chapter 2. Local Existence 34
where
kuk2H = max0�t�T
8<:Z
�u2t + l jruj2
�(x; t)dx
9=;Denote w = v� � v� for �; � given. Then w is a solution of the Cauchy problem:8>>>>>>>>>>>><>>>>>>>>>>>>:
wtt ��w ��wt + L(w) + k(v�t )� k(v�t )
= f(u�)� f(u�); x 2 ; t > 0
w(0; x) = u�0 � u�0; wt(0; x) = u�1 � u�1; x 2
w(t; x) = 0; x 2 �; t > 0
; (2.67)
where,
k(v�t ) = jv�t jm�2 v�t
f(u�) = ju�jp�2 u�
L(w) =
tZ0
g(t� s)�w(s; x)ds:
The energy equality reads as
1
2
d
dt
8<:kwtk22 +0@1� tZ
0
g(s)ds
1A krwk22 + (g � rw)(t)9=;
+
Z
�k(v�t )� k(v�t )
�wtdx+ krwtk22
(2.68)
=
Z
�f(u�)� f(u�)
�wtdx+
1
2(g0 � rw) (s)ds� 1
2krw(s)k22
tZ0
g(s)ds:
The term, Z
�k(v�t )� k(v�t )
�wtdx =
Z
�jv�t j
m�2 v�t ����v�t ���m�2 v�t��v�t � v�t
�dx
is nonnegative.
We need to estimate������Z
(f(u)� f(u)) (v � v) dx
������ � C(kukH + kukH)p�2 ku� ukH kv � vkH ;
Section 2.1. Local Existence Result
Chapter 2. Local Existence 35
ful�lled for u; u; v; v 2 H10 (); where C is a constant depending on ; l; p only. Then, Hölder�s
inequality yields, for1
q+1
n+1
2= 1 (q =
2n
n� 2);������Z
�f(u�)� f(u�)
�wtdx
������ =
������Z
�ju�jp�2 u� �
��u���p�2 u���v�t � v�t
�dx
������� C
u� � u� Lq
v�t � v�t
L2
�ku�kp�2n(p�2) +
u� p�2n(p�2)
�:
(2.69)
The Sobolev embedding Lq ,! H10 () gives u� � u�
Lq()
� C ru� �ru�
L2():
Then,
ku�kp�2n(p�2) + u� p�2
n(p�2) � C(ku�kp�2L2() + u� p�2
L2()):
The necessity to estimate ku�kn(p�2) by the energy norm kukH requires a restriction on p. Namely,
we need n(p� 2) � 2n
n� 2 ; then the Sobolev embedding Lq ,! H1
0 () gives
ku�kp�2n(p�2) � kukp�2H :
Therefore, (2:69) takes the form������Z
�f(u�)� f(u�)
�wtdx
������� C
v�t � v�t
L2()
ru� �ru� L2()
�kru�kp�2L2() +
ru� p�2L2()
�(2.70)
under the fact thattZ0
(g0 � rw) (s)ds � 0;
we conclude
�tZ0
(g0 � rw) (s)ds+ (g � rw) (t) + krw(t)k22
tZ0
g(s)ds � 0:
Thus,
1
2kw(t; :)k2H �
1
2kw(0; :)k2H + C
tZ0
ru� �ru� L2()
kwt(s; :)kH ds:
The Gronwell Lemma and Young�s inequality guarantee that
kw(t; :)kH � kw(0; :)kH + CT u� � u�
C([0;T ];H)
:
Section 2.1. Local Existence Result
Chapter 2. Local Existence 36
Since v�(t; :)� v�(t; :) H� C
v�(0; :)� v�(0; :) H+ CT
u� � u� C([0;T ];H)
; (2.71)
then fv�g is a Cauchy sequence in C([0; t] ; H); since fu�g and fv�(0; :)g are Cauchy sequences inC([0; T ] ; H) and H, respectively.
Now, we shall prove that fv�t g is a Cauchy sequence, in Lm ([0; T ]� ), to control the normkv�t k
2Lm([0;T ]�) : By the following algebraic inequality�
� j�jm�2 � � j�jm�2�(�� �) � C j�� �jm ; (2.72)
which holds for any real �; � and c, we getZ
�k(v�t )� k(v�t )
�wtdx =
Z
�v�t jv�t j
m�2 � v�t
���v�t ���m�2��v�t � v�t
�dx
� C���v�t � v�t
���2Lm([0;T ]�)
:
This estimate combined with (2:68) gives v�t � v�t
2Lm([0;t]�)
� C v�(0; :)� v�(0; :)
Lm([0;t]�)
+CR
tZ0
u� � u� Lm([0;t]�)
v�(t; :)� v�(t; :) Lm([0;t]�) ds:
So by using Gromwell Lemma, we obtain fv�t g is a Cauchy sequence, in Lm ([0; T ]� ) and hencefv�g is a Cauchy sequence in YT : Let v its limit in YT and by Lemma 2.1.2, v is a weak solution of(2:1).
Now, we are ready to show the local existence of the problem (P )
Proof of Theorem 2.1.1.
Let (u0; u1) 2 (H10 ())
2; and
R2 =�kru0k22 + ku1k
22
�:
For any T > 0; consider
MT =�u 2 YT : u(0) = u0; ut(0) = u1 and kukYT � R
:
Let� : MT !MT
u 7! v = �(u):
Section 2.1. Local Existence Result
Chapter 2. Local Existence 37
We will prove as in [13] that,
(i) �(MT ) �MT :
(ii) � is contraction in MT :
Beginning by the �rst assertion. By Lemma 2.1.1, for any u 2 MT we may de�ne v = �(u);
the unique solution of problem (2:1). We claim that, for a suitable T > 0, � is contractive map
satisfying
�(MT ) �MT :
Let u 2MT ; the corresponding solution v = �(u) satis�es for all t 2 [0; T ] the energy identity :
1
2
8<:kv0(t)k22 +0@1� tZ
0
g(s)ds
1A krv(t)k22 + (g � rv)(t)9=;
+
tZ0
krv0(s)k22 ds+tZ0
kv0(s)kmm)ds
(2.73)
=1
2
�kv1k22 + krv0k
22
�+
tZ0
Z
ju(s)jp�2 u(s)v0(t)dxds:
We get
1
2kv(t)k2YT �
1
2kv(0)k2YT +
tZ0
Z
ju(s)jp�2 u(s)v0(t)dxds: (2.74)
We estimate the last term in the right-hand side in (2:74) as follows: thanks to Hölder�s, Young�s
inequalities, we have Z
ju(s)jp�2 u(s)v0(t)dx � C kukpYT kvkYT ;
then,
kv(t)k2YT � kv(0)k2YT+ CRp
tZ0
kvkYT ds;
where C depending only on T; R: Recalling that u0; u1 converge, then
kv(t)kYT � kv(0)kYT + CRpT:
Choosing T su¢ ciently small, we getkvkYT � R; which shows that
�(MT ) �MT :
Section 2.1. Local Existence Result
Chapter 2. Local Existence 38
Now, we prove that � is contraction inMT : Taking w1 and w2 inMT , subtracting the two equations
in (2:1); for v1 = �(w1) and v2 = �(w2); and setting v = v1 � v2; we obtain for all � 2 H10 () and
a.e. t 2 [0; T ]
Z
vtt:�dx+
Z
rvr�dx+Z
rvtr�dx+Z
tZ0
g(t� s)rvr�dsdx
+
Z
�jvtjm�2 vt
��dx
=
Z
�jw1jp�2w1 � jw2jp�2w2
��dx: (2.75)
Therefore, by taking � = vt in (2:75) and using the same techniques as above, we obtain
kv(t; :)k2YT � C
tZ0
�kw1kp�2YT
+ kw2kp�2YT
�kw1 � w2kYT kv(s; :)k
2YTds: (2.76)
It�s easy to see that
kv(t; :)k2Yt = k�(w1)� �(w2)k2Yt� � kw1 � w2k2Yt ; (2.77)
for some 0 < � < 1 where � = 2CTRP�2:
Finally by the contraction mapping theorem together with (2:77); we obtain that there exists a
unique weak solution u of u = �(u) and as �(u) 2 YT we have u 2 YT : So there exists a unique
weak solution u to our problem (P ) de�ned on [0; T ] ; The main statement of Theorem 2.1.1 is
proved.
Remark 2.1.1 Let us mention that in our problem (P ) the existence of the term source ( f(u) =
jujp�2 u ) in the right hand forces us to use the contraction mapping theorem. Since we assume alittle restriction on the initial data. To this end, let us mention again that our result holds by the
well depth method, by choosing the initial data satisfying a more restrictions.
Section 2.1. Local Existence Result
Chapter 3
Global Existence and Energy Decay
Abstract
In this chapter, we prove that the solution obtained in the second chapter (Local solution ) is
global in time: In addition, we show that the energy of solutions decays exponentially if m = 2 and
polynomial if m > 2, provided that the initial data are small enough. The existence of the source
term�jujp�2 u
�forces us to use the potential well depth method in which the concept of so-called
stable set appears. We will make use of arguments in [44] with the necessary modi�cations imposed
by the nature of our problem.
39
Chapter 3. Global Existence and Energy Decay 40
3.1 Global Existence Result
In order to state and prove our results, we introduce the functional
I(t) = I(u(t)) =
0@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)� b ku(t)kpp ; (3.1)
and
J(t) = J(u(t)) =1
2
0@1� tZ0
g(s)ds
1A kru(t)k22 + 12(g � ru)(t)� b
pku(t)kpp ; (3.2)
for u(t; x) 2 H10 () ; t � 0:
As in [19], the potential well depth, is de�ned as
d = infu2H1
0 ()�f0gsup��0
J (�u) : (3.3)
The functional energy associated to (P ) is de�ned as fallows
E(u(t); ut(t)) = E(t) =1
2kut(t)k22 + J(t): (3.4)
Now, we introduce the stable set as follows:
W =�u 2 H1
0 () : J(u) < d; I(u) > 0[ f0g : (3.5)
We will prove the invariance of the set W: That is if for some t0 > 0 if u(t0) 2 W; then u(t) 2 W;
8t � t0:
Lemma 3.1.1 d is positive constant.
Proof. We have
J(�u) =�2
2
0@0@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)1A� b
p�p ku(t)kpp : (3.6)
Using (G1); (G2) to get
J(�u) � K(�);
where K(�) =�2
2l kruk22 �
b
p�p kukpp :
Section 3.1. Global Existence Result
Chapter 3. Global Existence and Energy Decay 41
By di¤erentiating the second term in the last equality with respect to �; to get
d
d�K(�) = �l kruk22 � b�p�1 kukpp : (3.7)
For, �1 = 0 and �2 =
l kruk22b kukpp
! 1p�2
; then we have
d
d�K(�) = 0:
As K(�1) = 0, we have
K(�2) =1
2
l kruk22b kukpp
! 2p�2
l kruk22 �b
p
l kruk22b kukpp
! pp�2
kukpp
=1
2b�2p�2 (l)
pp�2�kukpp
� �2p�2 �kruk22� p
p�2
�1pb�2p�2 (l)
pp�2�kukpp
�� 2p�2 �kruk22� p
p�2
= (l)p
P�2 b�2p�2�1
2� 1p
�kruk
2pp�22 kuk
�2pp�2p : (3.8)
By Sobolev-Poincaré�s inequality, we deduce that K(�2) > 0: Then, we obtain
sup fJ(�u); � � 0g � sup fK(�); � � 0g
> 0: (3.9)
Then, by the de�nition of d; we conclude that d > 0:
Lemma 3.1.2 ([19]) W is a bounded neighbourhood of 0 in H10 ().
Proof. For u 2 W; and u 6= 0; we have
J(t) =1
2
0@1� tZ0
g(s)ds
1A kru(t)k22 + 12(g � ru)(t)� b
pku(t)kpp
=
�p� 22p
�240@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)35+ 1
pI(u(t))
��p� 22p
�240@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)35 : (3.10)
Section 3.1. Global Existence Result
Chapter 3. Global Existence and Energy Decay 42
By using (G1) and (G2) then (3:10) becomes
J(t) ��p� 22p
�0@1� tZ0
g(s)ds
1A kru(t)k22� l
�p� 22p
�kru(t)k22
then,
kru(t)k22 � 1
l
�2p
p� 2
�J(t)
<1
l
�2p
p� 2
�d = R:
Consequently, 8u 2 W we have u 2 B where
B =�u 2 H1
0 () : kru(t)k22 < R
: (3.11)
This completes the proof.
Now, we will show that our local solution u(t; x) is global in time, for this purpose it su¢ ces to
prove that the norm of the solution is bounded, independently of t; this is equivalent to prove the
following theorem.
Theorem 3.1.1 Suppose that (G1) ; (G2) and (2:3) hold. if u0 2 W; u1 2 H10 () and
bCp�l
�2p
(p� 2) lE(0)�p�2
2
< 1; (3.12)
where C� is the best Poincaré�s constant. Then the local solution u(t; x) is global in time.
Remark 3.1.1 Let us remark, that if there exists t0 2 [0; T ) such that u(t0) 2 W and ut(t0) 2H10 () and condition (3:12) holds for t0: Then the same result of theorem 3.1.1 stays true.
Before we prove our results, we need the following Lemma, which means that, our energy is uniformly
bounded and decreasing along the trajectories.
Lemma 3.1.3 ([44]) Suppose that (G1) ; (G2), (2:3) hold, and let (u0; u1) 2 (H10 ())
2. Let u(t; x)
be the solution of (P ) ; then the modi�ed energy E(t) is non-increasing function for almost every
t 2 [0; T ) ; andd
dtE(t) = �a kut(t)kmm +
1
2(g0 � ru) (t)� 1
2g(t) kru(t)k22 � w krut(t)k22
(3.13)
� 0; 8t 2 [0; T ) :
Section 3.1. Global Existence Result
Chapter 3. Global Existence and Energy Decay 43
Proof. By multiplying the di¤erential equation in (P ) by ut and integration over we obtain
d
dt
8<:12 kut(t)k22 + 120@1� tZ
0
g(s)ds
1A kru(t)k22 + 12(g � ru)(t)� b
pku(t)kpp
9=;= �a kut(t)kmm +
1
2(g0 � ru) (t)� 1
2g(t) kru(t)k22 � w krut(t)k22
� 1
2(g0 � ru) (t) � 0; 8t 2 [0; T ) :
By the de�nition of E(t), we conclude
d
dtE(t) � 0: (3.14)
This completes the proof.
The following lemma tells us that if the initial data ( or for some t0 > 0) is in the sat W , then the
solution stays there forever.
Lemma 3.1.4 ([44]) Suppose that (G1) ; (G2) ; (2:3) and (3:12) hold. If u0 2 W; u1 2 H10 (),
then the solution u(t) 2 W; 8t � 0.
Proof. Since u0 2 W , then
I(0) = kru0k22 � ku0kpp > 0;
consequently, by continuity, there exists Tm � T such that
I(u (t)) =
0@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)� b ku(t)kpp � 0; 8t 2 [0; Tm] :
This gives
J(t) =1
2
0@1� tZ0
g(s)ds
1A kru(t)k22 + 12(g � ru)(t)� b
pku(t)kpp
=
�p� 22p
�240@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)35+ 1
pI(u(t))
��p� 22p
�240@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)35 : (3.15)
Section 3.1. Global Existence Result
Chapter 3. Global Existence and Energy Decay 44
By using (3:1) ; (3:15) and the fact thattR0
g(s)ds �1R0
g(s)ds; we easily see that
kru(t)k22 � 1
l
�2p
p� 2
�J(t)
� 1
l
�2p
p� 2
�E(u(t)) (3.16)
� 1
l
�2p
p� 2
�E(0); 8t 2 [0; Tm] :
We then exploit (G1) ; (3:12) ; (3:16) ; and we note that the embedding H10 () ,! Lp (), we have
ku(t)kp � C kru(t)k2 (3.17)
for 2 < p � 2n
n� 2 if n � 3, or p > 2 if n = 1; 2; and C = C (n; p; ) :
Consequently, we have
b ku(t)kpp � bCp� kru(t)kp2 ; 8t 2 [0; Tm]
� bCp� kru(t)kp�22 kru(t)k22
� bCp�lkru(t)kp�22 l kru(t)k22
� �l kru(t)k22 ;
(3.18)
which means by the de�nition of l
b ku(t)kpp � �
0@1� tZ0
g(s)ds
1A kru(t)k22<
0@1� tZ0
g(s)ds
1A kru(t)k22 ; 8t 2 [0; Tm] :
where
� =bCp�l
�2p
(p� 2) lE(0)�p�2
2
: (3.19)
Therefore,
I(t) =
0@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)� b ku(t)kpp > 0: (3.20)
Section 3.1. Global Existence Result
Chapter 3. Global Existence and Energy Decay 45
for all t 2 [0; Tm] ;By taking the fact that
limt7!Tm
bCp�l
�2p
(p� 2) lE(0)�p�2
2
� � < 1: (3.21)
This shows that the solution u(t) 2 W; for all t 2 [0; Tm] : By repeating this procedure Tm extendedto T:
Proof of Theorem 3.1.1.
In order to prove theorem 3.1.1, it su¢ ces to show that the following norm
kru(t)k2 + kut(t)k2 ; (3.22)
is bounded independently of t.
To achieve this, we use (3:4) ; (3:14) and (3:15) to get
E(0) � E(t) = J(t) +1
2kut(t)k22
��p� 22p
�240@1� tZ0
g(s)ds
1A kru(t)k22 + (g � ru)(t)35
+1
2kut(t)k22 +
1
pI(t)
��p� 22p
��l kru(t)k22 + (g � ru)(t)
�+1
2kut(t)k22 +
1
pI(t)
(3.23)
��p� 22p
��l kru(t)k22 +
1
2kut(t)k22
�;
since I(t) and (g � ru)(t) are positive, hence
kru(t)k22 + kut(t)k22 � CE(0);
where C is a positive constant depending only on p and l.
This completes the proof of theorem 3.1.1.
Section 3.1. Global Existence Result
Chapter 3. Global Existence and Energy Decay 46
The following lemma is very useful
Lemma 3.1.5 ([44]) Suppose that (2:3) and (3:12) hold. Then
b ku(t)kpp � (1� �)
0@1� tZ0
g(s)ds
1A kru(t)k22 (3.24)
where � = 1� �:
3.2 Decay of Solutions
We can now state the asymptotic behavior of the solution of (P ).
Theorem 3.2.1 Suppose that (G1) ; (G2) and (2:3) hold. Assume further that u0 2 W and u1 2H10 () satisfying (3:12) : Then the global solution satis�es
E(t) � E (0) exp (��t) ; 8t � 0 if m = 2; (3.25)
or
E(t) ��E(0)�r +K0rt
�s; 8t � 0 if m > 2, (3.26)
where � and K0 are constants independent of t; r =m
2� 1 and s = 2
2�m:
The following Lemma will play a decisive role in the proof of our result. The proof of this lemma
was given in Nakao [34].
Lemma 3.2.1 ( [37]) Let '(t) be a nonincreasing and nonnegative function de�ned on [0; T ] ;
T > 1; satisfying
'1+r(t) � k0 (' (t)� ' (t+ 1)) ; t 2 [0; T ] ;
for k0 > 1 and r � 0: Then we have, for each t 2 [0; T ] ;8>><>>:' (t) � ' (0) exp
��k [t� 1]+
�; r = 0
' (t) ��' (0)�r + k0r [t� 1]+
�1r ; r > 0
; (3.27)
where [t� 1]+ = max ft� 1; 0g ; and k = ln�
k0k0 � 1
�:
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 47
Proof of Theorem 3.2.1.
Multiplying the �rst equation in (P ) ; by ut and integrate over ; to obtain
d
dtE(t) + w krutk22 + a kutkmm =
1
2(g0 � ru) (t)� 1
2g(t) kru(t)k22 :
Then, integrate the last equality over [t; t+ 1] to get
E(t+ 1)� E(t) + w
t+1Zt
krutk22 ds+ a
t+1Zt
kutkmm ds
=
t+1Zt
1
2(g0 � ru) (s)ds�
t+1Zt
1
2g(s) kru(t)k22 ds:
Therefore,
E(t)� E(t+ 1) = Fm(t)� 12
t+1Zt
(g0 � ru) (s)ds+ 12
t+1Zt
g(t) kru(t)k22 ds; (3.28)
where
Fm(t) = a
t+1Zt
kutkmm ds+ w
t+1Zt
krutk22 ds: (3.29)
Using Poincaré�s inequality to �nd
t+1Zt
kutk22 ds � C ()
t+1Zt
kutk2m ds: (3.30)
Exploiting Hölder�s inequality, we obtain
t+1Zt
kutk2m ds �
0@ t+1Zt
ds
1Am�2m0@ t+1Z
t
�kutk2m
�m2
1A2m
ds
�
0@ t+1Zt
�kutk2m
�m2
1A2m
ds: (3.31)
Combining (3:29); (3:30); and (3:31); we obtain, for a constant C1; depending on
t+1Zt
kutk22 ds � C1F2(t); C1 > 0: (3.32)
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 48
By applying the mean value theorem, ( Theorem 1.3.3. in chapter1), we get for some t1 2�t; t+
1
4
�;
t2 2�t+
3
4; t+ 1
�kut(ti)k2 � 2C ()
12 F (t); i = 1; 2: (3.33)
Hence, by (G2) and sincet+1Zt
krutk22 ds � C2F (t)2; C2 > 0 (3.34)
there exist t1 2�t; t+
1
4
�; t2 2
�t+
3
4; t+ 1
�such that
krut(ti)k22 � 4C ()F (t)2; i = 1; 2: (3.35)
Next, we multiply the �rst equation in (P ) by u and integrate over � [t1; t2] to obtaint2Zt1
240@1� tZ0
g(�)d�
1A kru(t)k22 ds� b kukpp
35 ds= �
t2Zt1
Z
u:uttdxds� w
t2Zt1
Z
ru:rutdxds� a
t2Zt1
Z
u: jutjm�2 utdxds
+
t2Zt1
sZ0
g(s� �)
Z
ru(s): [ru(�)�ru(s)] dxd�ds.
Obviously,t2Zt1
I(s)ds = �t2Zt1
Z
u:uttdxds� w
t2Zt1
Z
ru:rutdxds� a
t2Zt1
Z
u: jutjm�2 utdxds
+
t2Zt1
sZ0
g(s� �)
Z
ru(s): [ru(�)�ru(s)] dxd�ds
+
t2Zt1
(g � ru) (s)ds: (3.36)
Note that by integrating by parts, to obtain������t2Zt1
Z
u:uttdxds
������ =
������24Z
utudx
35t2t1
�t2Zt1
Z
u2tdxds
������=
������Z
ut(t2)u(t2)dx�Z
ut(t1)u(t1)dx�t2Zt1
kutk22 ds
������ ;Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 49
Using Hölder�s and Poincaré�s inequalities, we get������t2Zt1
Z
u:uttdxds
������ � C2�
2Xi=1
krut(ti)k2 kru(ti)k2 + C2�
t2Zt1
krutk22 dt: (3.37)
By using Hölder�s inequality once again, we have������t2Zt1
Z
ru:rutdxds
������ �t2Zt1
kruk2 krutk2 ds (3.38)
Furthermore, by (3:35) and (3:16), we have
krut(ti)k2 kru(ti)k2 � C3 (C())12 F (t) sup
t1�s�t2E(s)
12 ; (3.39)
where, C3 = 2�
2p
l (p� 2)
�12
:
From (3:34) we have by Hölder�s inequality
t2Zt1
kruk2 krutk2 dt �t2Zt1
E(s)12
�1
l
�2p
p� 2
��12
krutk2 ds
� 1
2C3 sup
t1�s�t2E(s)
12
t2Zt1
krutk2 ds;
which implies
t2Zt1
krutk2 dt �
0@ t2Zt1
1dt
1A120@ t2Zt1
krutk22 dt
1A12
�p3C22
F (t):
Then,t2Zt1
kruk2 krutk2 dt � C4F (t) supt1�s�t2
E(s)12 (3.40)
where C4 =C3p3C24
: Therefore (3:37) ; becomes������t2Zt1
Z
u:uttdxds
������ � 2C2�C3F (t) supt1�s�t2E(s)
12 + C2�C2F (t)
2: (3.41)
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 50
We then exploit Young�s inequality to estimate
t2Zt1
Z
tZ0
g(s� �)ru(t): [ru(s)�ru(t)] d�dxdt
(3.42)
� �
t2Zt1
tZ0
g(s� �) kruk22 d�dt+1
4�
t2Zt1
(g � ru) (t)dt; 8� > 0:
Now, the third term in the right-hand side of (3:36), can be estimated as follows
t2Zt1
Z
jutjm�2 ut:udxds �t2Zt1
Z
jutjm�1 : juj dxds:
By Hölder�s inequality, we �nd
t2Zt1
Z
jutjm�1 : juj dxds �t2Zt1
26640@Z
jutjm dx
1Am�1m0@Z
jujm dx
1A1m
3775 ds
=
t2Zt1
kutkm�1m kukm ds:
By Sobolev-Poincaré�s inequality, we have
t2Zt1
kutkm�1m kukm ds � C ()
t2Zt1
kutkm�1m kruk2 ds;
for 2 < m � 2n
n� 2 if n � 3, or 2 � m <1 if n = 1; 2:
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 51
Using Hölder�s inequality, and since t1; t2 2 [t; t+ 1] and E(t) decreasing in time, we conclude fromthe last inequality, (3:16) and (3:29) ; that
t2Zt1
kutkm�1m kukm ds � C ()
�2p
l (p� 2)
�12
t2Zt1
kutkm�1m (J(u))12 ds
� C ()
�2p
l (p� 2)
�12
t2Zt1
kutkm�1m (E(u))12 ds
� C ()
�2p
l (p� 2)
�12
supt1�s�t2
(E(u))12 �
0@ t2Zt1
kutkmm ds
1Am�1m0@ t2Zt1
ds
1A1m
(3.43)
��1
a
�m�1m
C () supt1�s�t2
(E(t))12
�2p
l (p� 2)
�12
F (t)m�1
Then, taking into account (3:41)� (3:43), estimate (3:36) takes the form
t2Zt1
I(t)dt ��2C2� +
p3C24
w
�C3F (t) sup
t1�s�t2E(s)
12 + C2�C2F (t)
2
+a1m
2C3C () sup
t1�s�t2(E(t))
12 F (t)m�1
(3.44)
+�
t2Zt1
tZ0
g(t� s) kruk22 dsdt+�1
4�+ 1
� t2Zt1
(g � ru) (t)dt:
Moreover, from (3:4) and (3:10), we see that
E(t) =1
2kutk22 dt+ J(t)
=1
2kutk22 +
�p� 22p
�0@1� tZ0
g(s)ds
1A kruk22+
�p� 22p
�(g � ru) (t) + 1
pI(t): (3.45)
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 52
By integrating (3:45) over [t1; t2] ; we obtain
t2Zt1
E(t)dt =1
2
t2Zt1
kutk22 dt+�p� 22p
� t2Zt1
0@1� tZ0
g(s)ds
1A kruk22 dt+
�p� 22p
� t2Zt1
(g � ru) (t)dt+ 1p
t2Zt1
I(t)dt; (3.46)
which implies by exploiting (3:32)
t2Zt1
E(t)dt � C12(F (t))2 +
1
p
t2Zt1
I(t)dt+
�p� 22p
� t2Zt1
(g � ru) (t)dt
(3.47)
+
�p� 22p
� t2Zt1
0@1� tZ0
g(s)ds
1A kruk22 dt:By using (3:11), Lemma 3.1.5, we see that0@1� tZ
0
g(s)ds
1A kruk22 � 1
�I(t): (3.48)
Therefore, (3:47), takes the form
t2Zt1
E(t)dt � C ()
2(F (t))2 +
�p� 22p
� t2Zt1
(g � ru) (t)dt
+
�1
p+p� 22p�
� t2Zt1
I(t)dt: (3.49)
Again an integration of (3:14) over [s; t2] ; s 2 [0; t2] gives
E(s) = E(t2) + a
t2Zs
kut(t)kmm d� +1
2
t2Zs
g(�) kru(t)k22 d�
�12
t2Zs
�g0 � ru
�(t)d� + w
t2Zs
krut(t)k22 d� (3.50)
By using the fact that t2 � t1 �1
2, we have
t2Zt1
E(s)ds �t2Zt1
E(t2)ds �1
2E(t2): (3.51)
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 53
The fourth term in (3:44), can be handled as
tZ0
g(t� s) kruk22 ds = kruk22
tZ0
g(t� s)ds
(3.52)
� 2p (1� l)
l (p� 2) E(t):
Thus,
t2Zt1
tZ0
g(t� s) kruk22 dsdt � 2p (1� l)
l (p� 2)
t2Zt1
E(t)dt
� p (1� l)
l (p� 2)E(t1)
� p (1� l)
l (p� 2)E(t): (3.53)
Hence, by (3:53) ; we obtain from (3:44)
t2Zt1
I(t)dt ��2C2� +
p3C24
w
�C3F (t) sup
t1�s�t2E(s)
12 + C2�C2F (t)
2
+a1m
2C3C () sup
t1�s�t2(E(t))
12 F (t)m�1
+�p (1� l)
l (p� 2)E(t) +�1
4�+ 1
� t2Zt1
(g � ru) (t)dt: (3.54)
From (3:50) and (3:51) we have
E(t) � 2
t2Zt1
E(s)ds+ a
t+1Zt
kut(t)kmm d� +1
2
t+1Zt
g(�) kru(t)k22 d�
�12
t+1Zt
(g0 � ru) (t)d� + w
t+1Zt
krut(t)k22 d� : (3.55)
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 54
Obviously, (3:49) and (3:55) give us
E(t) � 2
0@C ()2
(F (t))2 +
�p� 22p
� t2Zt1
(g � ru) (t)dt+�1
p+p� 22p�
� t2Zt1
I(t)dt
1A+a
t2Zt1
kut(t)kmm dt+1
2
t2Zt1
g(t) kru(t)k22 dt�1
2
t2Zt1
(g0 � ru) (t)dt
+w
t2Zt1
krut(t)k22 dt:
Consequently, plugging the estimate (3:54) into the above estimate, we conclude
E(t) � C () (F (t))2 +
�p� 2p
� t2Zt1
(g � ru) (t)dt
+2
�1
p+p� 22p�
���2C2� +
p3C24
w
�C3F (t) sup
t1�s�t2E(s)
12 + C2�C2F (t)
2
�
+
�1
p+p� 22p�
�a1mC3C () sup
t1�s�t2(E(t))
12 F (t)m�1
+2
�1
p+p� 22p�
�24�p (1� l)
l (p� 2)E(t) +�1
4�+ 1
� t2Zt1
(g � ru) (t)dt)
35(3.56)
+Fm(t)� 12
t2Zt1
(g0 � ru) (t)dt+ 12
t2Zt1
g(t) kru(t)k22 dt:
We also have, by the Poincaré�s inequality
ku(s)k2 � C kru(s)k2
� C
�2p
l (p� 2)
�12
E(t)12 ; (3.57)
Choosing � small enough so that
1� 2�1
p+p� 22p�
��p (1� l)
l (p� 2) > 0; (3.58)
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 55
we deduce, from (3:56) that there exists K > 0 such that
E(t) � K
�F (t)2 + E(t)
12F (t) + E(t)
12F (t)m�1 + F (t)m
�
+1
2
t+1Zt
g(s) kru(s)k22 ds�1
2
t+1Zt
(g0 � ru) (s)ds
(3.59)
+
��p� 2p
�+ 2
�1
4�+ 1
��1
p+p� 22p�
�� t+1Zt
(g � ru) (s)ds
Using (G2) again we can write
t2Zt1
(g � ru) (t)dt � ��t2Zt1
(g0 � ru) (t)dt; � > 0:
Then, we obtain, from (3:59),
E(t) � K
�F (t)2 + E(t)
12F (t) + E(t)
12F (t)m�1 + F (t)m
�(3.60)
+1
2
t+1Zt
g(s) kru(s)k22 ds���1 +
1
2
� t+1Zt
(g0 � ru) (s)ds:
where �1 = �
��p� 2p
�+ 2
�1
p+p� 22p�
��1
4�+ 1
��:
An appropriate use of Young�s inequality in (3:60), we can �nd K1 > 0 such that
E(t) � K1
�F (t)2 + F (t)2(m�1) + F (t)m
�(3.61)
+
2412
t+1Zt
g(s) kru(s)k22 ds���1 +
1
2
� t+1Zt
(g0 � ru) (s)ds
35 ;for K1 a positive constant.
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 56
Using (G2) again to get
E(t) � K1
�F (t)2 + F (t)2(m�1) + F (t)m
�
+
24��1 + 12� t+1Z
t
g(s) kru(s)k22 ds���1 +
1
2
� t+1Zt
(g0 � ru) (s)ds
35� K1
�F (t)2 + F (t)2(m�1) + F (t)m
�(3.62)
+(1 + 2�1)
2412
t+1Zt
g(s) kru(s)k22 ds�1
2
t+1Zt
(g0 � ru) (s)ds
35At this end we distinguish two cases:
Case 1. m = 2. In this case we use (3:28) and (3:62), we can �nd K2 > 0 such that
E(t) � K1 F (t)2
+(1 + 2�1)
2412
t+1Zt
g(s) kru(s)k22 ds�1
2
t+1Zt
(g0 � ru) (s)ds
35� K2 [E(t)� E(t+ 1)] : (3.63)
Since E(t) is nonincreasing and nonnegative function, an application of Lemma 3.2.1 yields
E(t) � K2 [E(t)� E(t+ 1)] ; t � 0; (3.64)
which implies that
E(t) � E (0) exp��� [t� 1]+
�; on [0;1) ; (3.65)
where � = ln�
K2
K2 � 1
�:
Case 2. m > 2: In this case we, again use (3:28) and (3:62) to arrive at
F (t)2 =
24(E(t)� E(t+ 1))� 12
t+1Zt
g(s) kru(s)k22 ds+1
2
t+1Zt
(g0 � ru) (s)ds
352m
: (3.66)
We then use the algebraic inequality
(a+ b)m2 � 2
m2
�am2 + b
m2
�; m � 2: (3.67)
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 57
To infer from (3:62) ; and by using (3:67) ; that
[E(t)]m2 � K3
�1 + F (t)2(m�2) + F (t)m�2
�m2 F (t)m
+2m2 (1 + 2�1)
m2
2412
t+1Zt
g(s) kru(s)k22 ds�1
2
t+1Zt
(g0 � ru) (s)ds
35m2
� K3
�1 + F (t)2(m�2) + F (t)m�2
�m2 � [E(t)� E(t+ 1)]
(3.68)
+2m2 (1 + 2�1)
m2
2412
t+1Zt
g(s) kru(s)k22 ds�1
2
t+1Zt
(g0 � ru) (s)ds
35m2
where K3 = 2m2 K1: We use (3:28) to obtain0@1
2
t+1Zt
g(s) kruk22 ds�1
2
t+1Zt
(g0 � ru) (s)ds
1Am2
� (E(t)� E(t+ 1))m2 (3.69)
A combination of (3:68) ; (3:69) yields
[E(t)]m2 � K3
�1 + F (t)2(m�2) + F (t)m�2
�m2 � (E(t)� E(t+ 1))
+2m2 (1 + 2�1)
m2 [E(t)� E(t+ 1)]
m2�1 [E(t)� E(t+ 1)]
��K3
�1 + F (t)2(m�2) + F (t)m�2
�m2 + 2
m2 (1 + 2�1)
m2 [E(t)� E(t+ 1)]
m2�1��
[E(t)� E(t+ 1)] (3.70)
By using (3:62) ; the estimate (3:70) takes the form
[E(t)]m2 �
nK32
mh1 + E(0)(m�2) + (E(0))
m2�1i+ 2
m2 (1 + 2�1)
m2 (E(0))
m2�1o�
(E(t)� E(t+ 1))
� K0 (E(t)� E(t+ 1)) : (3.71)
Section 3.2. Decay of Solutions
Chapter 3. Global Existence and Energy Decay 58
Again, using Lemma 3.2.1, we conclude
E(t) ��E(0)�r +K0r [t� 1]+
�s; (3.72)
with r =m
2� 1 > 0; s = 2
2�mand K0 is some given positive constant.
This completes the proof.
Section 3.2. Decay of Solutions
Chapter 4
Exponential Growth
Abstract
Our goal in this chapter is to prove that when the initial energy is negative and p > m, then, the
solution with the Lp�norm grows as an exponential function provided that1R0
g(s)ds <p� 2p� 1 , by
using carefully the arguments of the method used in [16], with necessary modi�cation imposed by
the nature of our problem.
59
Chapter 4. Exponential Growth 60
4.1 Growth result
Our result reads as follows.
Theorem 4.1.1 Suppose that m � 2 and m < p � 1, if n = 1; 2; m < p � 2 (n� 1)n� 2 ; if n � 3:
Assume further that E(0) < 0 and1R0
g(s)ds <p� 2p� 1 holds: Then the unique local solution of problem
(P ) grows exponentially.
Proof. We set
H(t) = �E(t): (4.1)
By multiplying the �rst equations in (P ) by �ut, integrating over and using Lemma 2.1.3, weobtain
� d
dt
8<:12 kutk22 + 120@1� tZ
0
g(s)ds
1A kruk22 + 12 (g � ru) (t)� b
pkukpp
9=;= a kutkmm �
1
2(g0 � ru) (t) + 1
2g(t) kruk22 + w krutk22 : (4.2)
By the de�nition of H(t); (4:2) rewritten as
H 0(t) = a kutkmm �1
2(g0 � ru) (t) + 1
2g(t) kruk22 + w krutk22 � 0; 8t � 0: (4.3)
Consequently, E(0) < 0; we have
H(0) = �12ku1k22 �
1
2kru0k22 +
b
pku0kpp > 0: (4.4)
It�s clear that by (4:1), we have
H(0) � H(t); 8t � 0: (4.5)
Using (G2) ; to get
H(t)� b
pkukpp = �
2412kutk22 +
1
2
0@1� tZ0
g(s)ds
1A kruk22 + 12 (g � ru) (t)35
� 0; 8t � 0: (4.6)
One implies
0 < H(0) � H(t) � b
pkukpp : (4.7)
Section 4.1. Growth result
Chapter 4. Exponential Growth 61
Let us de�ne the functional
L(t) = H(t) + "
Z
utudx+ "w
2kruk22 : (4.8)
for " small to be chosen later.
By taking the time derivative of (4:8) ; we obtain
L0(t) = H 0(t) + "
Z
uutt(t; x)dx+ " kutk22 + "w
Z
rutrudx
=
�w krutk22 + a kutkmm �
1
2(g0 � ru) (t) + 1
2g(t) kruk22
�
+" kutk22 + "w
Z
rutrudx+ "
Z
uttudx: (4.9)
Using the �rst equations in (P ), to obtainZ
uuttdx = b kukpp � kruk22 � !
Z
rutrudx� a
Z
jutjm�2 utudx
+
Z
rutZ0
g(t� s)ru(s; x)dsdx: (4.10)
Inserting (4:10) into (4:9) to get
L0(t) = w krutk22 + a kutkmm �1
2(g0 � ru) (t) + 1
2g(t) kruk22
+" kutk22 � " kruk22 + "
tZ0
g(t� s)
Z
ru:ru(s)dxds
+"b kukpp � "a
Z
jutjm�2 utudx: (4.11)
By using (G2) ; the last equality takes the form
L0(t) � w krutk22 + a kutkmm + " kutk22 � " kruk22 + "b kukpp(4.12)
+"
tZ0
g(t� s)
Z
ru:ru(s)dxds� "a
Z
jutjm�2 utud�:
To estimate the last term in the right-hand side of (4:12) ; we use the following Young�s inequality
XY � �r
rXr +
��q
qY q; X; Y � 0; (4.13)
Section 4.1. Growth result
Chapter 4. Exponential Growth 62
for all � > 0 be chosen later;1
r+1
q= 1; with r = m and q =
m
m� 1 :So we have Z
jutjm�2 utudx �Z
jutjm�1 juj dx
� �m
mkukmm +
�m� 1m
��(
�mm�1) kutkmm ; 8t � 0: (4.14)
Therefore, the estimate (4:12) takes the form
L0(t) � w krutk22 + a kutkmm + " kutk22 � " kruk22 + "b kukpp
+"
tZ0
g(t� s)
Z
ru:ru(s)dxds
�"a�m
mkukmm � "a
�m� 1m
��
� �mm�1
�kutkmm
� w krutk22 + a kutkmm + " kutk22 � " kruk22 + "b kukpp
+" kruk22
tZ0
g(s)ds+ "
tZ0
g(t� s)
Z
ru (t) [ru (s)�ru (t)] dxds
�"a�m
mkukmm � "a
�m� 1m
��
� �mm�1
�kutkmm : (4.15)
Using Cauchy-Schwarz and Young�s inequalities to obtain
L0(t) � w krutk22 + a
�1� "
�m� 1m
��
� �mm�1
��kutkmm + " kutk22
�" kruk22 + "b kukpp + " kruk22
tZ0
g(s)ds
�"tZ0
g(t� s) kruk2 kru(s)�ru (t)k2 ds� "a�m
mkukmm
� w krutk22 + a
�1� "
�m� 1m
��
� �mm�1
��kutkmm + " kutk22 + "b kukpp
(4.16)
+"
0@12
tZ0
g(s)ds� 1
1A kruk22 � "1
2(g � ru(t)� "a
�m
mkukmm :
Section 4.1. Growth result
Chapter 4. Exponential Growth 63
Using assumptions to substitute for b kukpp : Hence, (4:16) becomes
L0(t) � w krutk22 + a
�1� "
�m� 1m
��
� �mm�1
��kutkmm + " kutk22
+"
0@pH(t) + p
2kutk22 +
p
2(g � ru) (t) + p
2
0@1� tZ0
g(s)ds
1A kruk221A
+"
0@12
tZ0
g(s)ds� 1
1A kruk22 � "1
2(g � ru(t)� "a
�m
mkukmm :
� w krutk22 + a
�1� "
�m� 1m
��
� �mm�1
��kutkmm + "
�1 +
p
2
�kutk22
(4.17)
+"a1 kruk22 + "a2(g � ru(t)� "a�m
mkukmm + "pH(t):
where a1 =�1� p
2
� 1R0
g(s)ds+
�p� 22
�> 0; a2 =
p� 12
> 0:
In order to undervalue L0(t) with terms of E(t) and since p > m, we have from the embedding
Lp () ,! Lm () ;
kukmm � C kukmp � C�kukpp
�mp; 8t � 0: (4.18)
for some positive constant C depending on only. Since 0 <m
p< 1; we use the algebraic inequality
zk � (z + 1) ��1 +
1
w
�(z + w) ; 8z � 0; 0 < k � 1; w > 0;
to �nd �kukpp
�mp � K
�kukpp +H(0)
�; 8t � 0; (4.19)
where K = 1 +1
H(0)> 0; then by (4:7) we have
kukmm � C
�1 +
b
p
�kukpp ; 8t � 0: (4.20)
Inserting (4:20) into (4:17), to get
L0(t) � w krutk22 + a
�1� "
�m� 1m
��
� �mm�1
��kutkmm + "
�1 +
p
2
�kutk22
(4.21)
+"a1 kruk22 + "a2(g � ru(t)� "C1 kukpp + "pH(t):
where C1 = aC�m
m
�1 +
b
p
�> 0:
Section 4.1. Growth result
Chapter 4. Exponential Growth 64
By using (4:1) and by the same statements as in [16], we have
2H(t) = �kutk22 � kruk22 +
tZ0
g(s)ds kruk22 � (g � ru) (t) +2b
pkukpp
(4.22)
� �kutk22 � kruk22 � (g � ru) (t) +
2b
pkukpp ; 8t � 0:
Adding and substituting the value 2a3H(t) from (4:21), and choosing � small enough such that
a3 < min fa1; a2g ; we obtain
L0(t) � w krutk22 + a
�1� "
�m� 1m
��
� �mm�1
��kutkmm
+"�1 +
p
2� a3
�kutk22 + " (a1 � a3) kruk22
+" (a2 � a3) (g � ru(t) + "
�2b
pa3 � C1
�kukpp
+" (p� 2a3)H(t): (4.23)
Now, once � is �xed, we can choose " small enough such that
1� "
�m� 1m
��
� �mm�1
�> 0; and L(0) > 0: (4.24)
Therefore, (4:23) takes the form
L0(t) � "�nH(t) + kutk22 + kruk
22 + (g � ru(t)) + kuk
pp
o; (4.25)
for some � > 0:
Now, using (G2), Young�s and Poincaré�s inequalities in (4:8) to get
L(t) � �1�H(t) + kutk22 + kruk
22
; (4.26)
for some �1 > 0: Since, H(t) > 0; we have from (4:1)
� 12kutk22 �
1
2
0@1� tZ0
g(s)ds
1A kruk22 � 12 (g � ru) (t) + b
pkukpp > 0; 8t � 0: (4.27)
Then,
1
2
0@1� tZ0
g(s)ds
1A kruk22 <b
pkukpp
<b
pkukpp + (g � ru) (t): (4.28)
Section 4.1. Growth result
Chapter 4. Exponential Growth 65
In the other hand, using (G1) ; to get
1
2(1� l) kruk22 � 1
2
0@1� tZ0
g(s)ds
1A kruk22<
b
pkukpp + (g � ru) (t): (4.29)
Consequently,
kruk22 <2b
pkukpp + 2 (g � ru) (t) + 2l kruk
22 ; b; l > 0: (4.30)
Inserting (4:30) into (4:26) ; to see that there exists a positive constant � such that
L(t) � �
�H(t) + kutk22 + kruk
22 + (g � ru) (t) +
b
pkukpp
�; 8t � 0: (4.31)
From inequalities (4:25) and (4:31) we obtain the di¤erential inequality
L0(t)
L(t)� �; for some � > 0; 8t � 0: (4.32)
Integration of (4:32) ; between 0 and t gives us
L(t) � L(0) exp (�t) ; 8t � 0; (4.33)
From (4:8) and for " small enough, we have
L(t) � H(t) � b
pkukpp : (4.34)
By (4:33) and (4:34) ; we have
kukpp � C exp (�t) ; C > 0; 8t � 0: (4.35)
Therefore, we conclude that the solution in the Lp�norm growths exponentially.
4.1. Growth result
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