on an extension of the kato-voigt perturbation...

TAIWANESE JOURNAL OF MATHEMATICSVol. 5, No. 1, pp. 169-191, March 2001

ON AN EXTENSION OF THE KATO-VOIGT PERTURBATIONTHEOREM FOR SUBSTOCHASTIC SEMIGROUPS

AND ITS APPLICATION

J. Banasiak¤

Abstract. The aim of this paper is to discuss some recent developments of theperturbation method introduced first by Kato for the Kolmogoroff equation andlater extended by Voigt and Arlotti to deal with a range of problems relatedto the solvability of the so-called Master Equation. The paper consists of twoparts. In the first we recall and unify some abstract results on generation ofsubstochastic semigroups. In the second we perform a detailed analysis ofan equation from the polymer degradation theory to demonstrate a number ofpossible generation cases.

1. INTRODUCTION

In many branches of the applied sciences we are interested in the evolution ofthe density function (t; ») 7! u(t; »), where t is the time and » 2 is an elementof some state space – it could be the velocity or energy in the kinetic theory, thenumber of bacteria in a sample in the population theory, mass in the coagulation-fragmentation theory. The function u is then interpreted as the probability (density)of finding an individual which at the time t has the property ». If we assume thatthe probability at time t is completely determined by the probability at an earliertime t0 (that is, we are dealing with a Markov process), then the time evolution ofu is described by the master equation (e.g., [11])

@tu(t; »)=

Z

[k(»0; »)u(t; »0) ¡ k(»; »0)u(t; »)]d¹»0

=

Z

k(»0; »)u(t; »0)d¹»0 ¡ u(t; »)

Z

k(»; »0)d¹»0;(1.1)

Received February 19, 2000; revised June 4, 2000.Communicated by S.-Y. Shaw.2000 Mathematics Subject Classification: 47D06, 34G10.Key words and phrases: Perturbation theory for semigroups, Substochastic semigroup, Fragmentationmodel.¤This work was supported by the University of Natal Research Fund and was lectured at the ICMAA2000, National Sun Yat-sen University, Taiwan.

169

170 J. Banasiak

where d¹» is an appropriate measure in the state space, and k(»0; ») is the transitionrate (probability density of transitions per unit time) of the change between the states»0 and » (that is, the probability of the change from »0 to » to occur in the timeinterval dt is approximately k(»0; »)d»dt).

An intrinsic property of the above process is that all the particles must beaccounted for or, in other words, the total number of particles doesn’t change:

Z

u(t; »)d¹» =

Z

u(0; »)d¹»;(1.2)

for any time t. Therefore from the physical point of view, the natural spaces forstudying such problems are L1 spaces (or l1 if » takes only discrete values). Equation(1.2) can be checked formally by integrating (1.1) and using the symmetry of thetransition rate.

The process discussed above (or a combination of such processes) may takeplace simultaneously with an evolution in the physical space so that in general wewill be concerned with evolution equations of the form

@tu = A0u + A1u + Bu;(1.3)

where A0 typically is the free streaming operator: A0u = ¡ v ¢rxu, or the diffu-sion operator, whereas A1u = ¡ mu with a nonnegative, measurable, and almosteverywhere finite function m, and B is an integral operator.

The first equation of this type, namely the Kolmogoroff equation, was firststudied in the semigroup theory framework by Kato in his seminal paper [6]. Hisresults were extended to a more abstract framework and extensively applied to arange of problems arising mainly in the kinetic theory by Voigt, Arlotti, Desch,Mokhtar-Karroubi, and recently by the author (see, e.g., [13, 1, 2, 9, 3, 4]).

An interesting feature of these problems is that the semigroup (G(t))t¸ 0 solvingthe Cauchy problem related to (1.3) is constructed by a limiting procedure whichdoesn’t allow to control the domain of the generator; in the general case one canonly prove that the generator T of (G(t))t¸ 0 is an extension of A0 + A1 + B. Inmost applications it is not enough, as typically we are interested in the cases whenthe solution semigroup (G(t))t¸ 0 is a transition (or stochastic) semigroup, that is,when kG(t)xk = kxk for all t ¸ 0 and x ¸ 0 (see also Remark 2.1). In a number ofapplications (see [6]) this could happen if and only if T = A0 + A1 + B; generallythe last equality is a sufficient condition for (G(t))t¸ 0 to be stochastic. Thus it isimportant to be able to determine when T = A0 + A1 + B. We shall present somesufficient conditions, which are modifications of those introduced in [2], and applythem to the fragmentation model [14, 8] which will be introduced in Section 4. Weshall see that for some values of the parameter the generator T of the semigroupis the closure of the operator appearing on the right-hand side of the equation (and

Kato-Voigt Perturbation Theorem 171

that this result is sharp, that is, T 6= A0 + A1 + B). On the other hand, in the lastsection of the paper we shall demonstrate that for a range of parameter values thegenerator T is a proper extension of A0 + A1 + B and that the semigroup generatedby T is not stochastic.

Acknowledgment. The author is grateful to Dr. W. Lamb for communicatingthe fragmentation model to him, and for a number of stimulating discussions.

The author also expresses sincere thanks for the referee, who suggested a sim-plification of a number of proofs, based on a more extensive use of the Miyadera –Voigt theory.

2. ABSTRACT GENERATION RESULTS

In this section, we shall formulate and provide proofs of several results whichare scattered in the literature [13, 1, 2], and which are very often considered onlyin a specific context.

Let ( ; ¹) be a measure space. By X we denote the Banach space L1( ; ¹)endowed with the standard norm k ¢k. For any subspace Z ½ X, by Z+ wedenote the cone of nonnegative (a.e.) elements of Z. Let (G(t))t¸ 0 be a stronglycontinuous semigroup on X. We say that (G(t))t¸ 0 is a substochastic semigroupif for each t ¸ 0, G(t) ¸ 0 and kG(t)k · 1. It is called a stochastic semigroup ifadditionally kG(t)fk = kfk for X+.

In accordance with the discussion in the introduction, we shall consider twolinear operators in X: A(= A0 + A1) and B, which have the following properties:

1. (A; D(A)) generates a substochastic semigroup denoted by (GA(t))t¸ 0,

2. D(B) ¾ D(A) and Bf ¸ 0 for any f 2 D(B)+,

3. for any f 2 D(A)+,

Z

(Af + Bf)d¹ · 0:(2.1)

Let us observe that the above list yields the following properties of the operatorsinvolved.

Lemma 2.1. (i) The operator B(¸I ¡ A)¡ 1 is a bounded positive operator inX.

(ii) If Af = ¡ mf for some a.e. positive measurable m; then for any f 2 D(A);

172 J. Banasiak

kBfk · kAfk:(2.2)

Proof. (i) Since A generates a contraction semigroup, the resolvent is well-defined for ¸ > 0. From the properties of B we see that B(¸I ¡ A)¡ 1 is definedon X and positive, and hence is bounded [12, Theorem II. 5.3].

(ii) From (2.1) we obtain for f 2 D(A)+,

kBfk =

Z

Bfd¹ =

Z

mfd¹ = kAfk:

Let us take arbitrary f 2 D(A), then also jf j 2 D(A), jBf j · Bjf j and

kBfk · kBjfjk =

Z

Bjfjd¹ =

Z

mjf jd¹ =

Z

jmf jd¹ = kAfk:

The next lemma is a minor generalization of Lemma 1.2 of [13], and is also moregeneral than Lemma 1 of [1].

Lemma 2.2. For any f 2 D(A); the function t 7! BGA(t)f is continuous and

tZ

0

kBGA(s)fkds · kfk ¡ kGA(t)fk:(2.3)

Proof. For f 2 D(A) it follows as in the proof of Lemma 1.2 (a) in [13] that

tZ

0

kBGA(s)fkds · kjf jk ¡ kGA(t)jf jk:

However, kjf jk = kfk and, since GA(t)jf j ¸ jGA(t)f j, we obtain ¡ kGA(t)jfjk ·¡ kGA(t)fk, so that the lemma is proved.

The following gives an abstract version of Theorem 4 of [1].

Theorem 2.1. Under the above assumptions; there exists a smallest substochas-tic semigroup (G(t))t¸ 0 generated by an extension T of A + B. This semigroupsatisfies the integral equation

G(t)f = GA(t)f +

tZ

0

G(t ¡ s)BGA(s)fds(2.4)


for any f 2 D(A) and t ¸ 0, and can be also obtained by the Phillips-Dysonexpansion

G(t)f =1X

n=0

Sn(t)f; f 2 X;(2.5)

where the iterates Sn(t) are defined through (2.7).

Proof. In [13] the author proved, generalizing the idea of Kato [6], that thesemigroup (G(t))t¸ 0 can be obtained as a strong limit of the semigroups (Gr(t))t¸ 0

generated by A+rB, when 0 < r ! 1¡ . Moreover, if r · r0, then Gr(t) · Gr0(t)for each t. For r < 1, the operator rB is a Miyadera perturbation of A with boundr, and therefore

Gr(t)f = GA(t)f + r

tZ

0

Gr(t ¡ s)BGA(s)fds:(2.6)

Since also rGr(t) · r0Gr0(t) if r · r0, using the Lebesgue monotone convergencetheorem (in L1( £ [0; t])) we ascertain that the right-hand side of (2.6) converges

to GA(t)f +tR0

G(t ¡ s)BGA(s)fds as r ! 1¡ for all f 2 X+, thus, by linearity,

for all f 2 X. This proves (2.6).

To obtain the Phillips-Dyson expansion of the semigroup (G(t))t¸ 0, we definerecursively the following operators:

S0(t)f = GA(t)f;

Sn(t)f =

tZ

0

Sn¡ 1(t ¡ s)BGA(s)fds; n > 0;(2.7)

for f 2 D(A) and t ¸ 0. It follows that the Dyson-Phillips iterates for the pertur-bation rB are given by rSn(t), thus, as in the Miyadera-Voigt theory, the familiesfSn(t)gt¸ 0 can be extended to strongly continuous families of bounded (contractive)positive operators in X . Moreover, for any 0 < r < 1 the semigroup (Gr(t))t¸ 0 isgiven by

Gr(t)f =1X

n=0

rnSn(t)f;

where the series is uniformly convergent on bounded t-intervals. Since Sn(t) arepositive, for f 2 X+ and 0 < r0 · r < 1 we have r

0nSn(t)f · rnSn(t)f , thus

174 J. Banasiak

by the monotone convergence theorem (in l1(X)) the series above is convergent to1P

n=0Sn(t)f in X. Thus for all f 2 X+ we have

G(t)f =

1X

n=0

Sn(t)f:

The extension by linearity gives (2.5).

Remark 2.1. The drawback of Theorem 2.1 is that it doesn’t provide any char-acterization of the domain of the generator T . The ideal situation would be, ofcourse, if D(T) = D(A) = D(A+ B). However, as we mentioned in the introduc-tion, the case when T = A + B is also physically acceptable since in such a casethe semigroup is still stochastic provided A + B is formally conservative, that is,the assumption (2.1) is replaced by

Z

(A + B)ud¹ = 0

for u 2 D(A + B) = D(A). If T = A + B, then for u 2 D(T ) there exists asequence (un)n2N of elements of D(A) such that un ! u and (A + B)un ! T uin X as n ! 1. Thus

Z

T ud¹ = limn!1

Z

(A + B)und¹ = 0:

This in turn shows that if 0 · ±u u 2 D(T), then 0 · u(t) = G(t)

±u u 2 D(T ) for

any t ¸ 0 and

d

dtku(t)k =

Z

du(t)

dtd¹ =

Z

T G(t)±u ud¹ = 0:

On the other hand, if T is a bigger extension of A + B, then the above propertymay not hold and there may be a loss of particles in the evolution. We shall seethat such a situation is possible in Proposition 5.2.

The next proposition is related to the above discussion. We note that for aparticular case of Kolmogoroff’s equation similar results, though through a differentmethod, have been proved in [6].

Proposition 2.1. Under the assumption of this section we have:


( i ) the generator T is characterized by

(I ¡ T)¡ 1f =1X

n=0

(I ¡ A)¡ 1[B(I ¡ A)¡ 1]nf(2.8)

for every f 2 X.

(ii) If1P

n=0[B(I ¡ A)¡ 1]nf converges for any f 2 X; then T = A + B.

(iii) If limn!1

k[B(I ¡ A)¡ 1]nfk = 0 for any f 2 X; then T = A + B.

Proof. (i) Taking the Laplace transform of (2.5) with ¸ = 1, we obtain

(I ¡ T )¡ 1f =1X

n=0

L(Sn(t)f)(1):(2.9)

Now, for f 2 D(A),

L(Sn(t)f)(1) = L(Sn¡ 1(t)f)(1)L(BGA(t)f)(1);

where we used Corollary C.17 of [5] (applicable by Lemma 2.2). The next steprequires some care as we don’t know whether B is a closed operator. However,using the boundedness of B(I ¡ A)¡ 1 we have

L(BGA(t)f)(1) = L(B(I ¡ A)¡ 1GA(t)(I ¡ A)f)(1) = B(I ¡ A)¡ 1f:

Therefore, for any f 2 D(A),

L(S0(t)f)(1) = (I ¡ A)¡ 1f;

L(Sn(t)f)(1) = L(Sn¡ 1(t))B(I ¡ A)¡ 1f = (I ¡ A)¡ 1[B(I ¡ A)¡ 1]nf;

and since L(Sn(t)f)(1) is a bounded operator, we can extend the above to the wholeof X. Combining this with (2.9), we obtain (2.8).

(ii) If the series converges in X, then we clearly have

(I ¡ T)¡ 1f =1X

n=0

(I ¡ A)¡ 1[B(I ¡ A)¡ 1]nf = (I ¡ A)¡ 11X

n=0

[B(I ¡ A)¡ 1]nf;

which shows that D(A) ¾ D(T). Since D(A) ½ D(T ), we have the equality.(iii) Assume that for any f 2 X, k(B(I ¡ A)¡ 1)nfk ! 0 as n ! 0. Denote

zN = (I ¡ A)¡ 1NX

n=0

(B(I ¡ A)¡ 1)nf 2 D(A):

176 J. Banasiak

Clearly, zN ! z = (I ¡ T )¡ 1f 2 D(T ) in X . Now,

(I ¡ (A + B))zN =

NX

n=0

(B(I ¡ A)¡ 1)nf ¡ B(I ¡ A)¡ 1NX

n=0

(B(I ¡ A)¡ 1)nf

=NX

n=0

(B(I ¡ A)¡ 1)nf ¡N+1X

n=1

(B(I ¡ A)¡ 1)nf

= f ¡ (B(I ¡ A)¡ 1)Nf:

Thus, if the assumption is satisfied, then (I ¡ (A + B))zN converges in X andconsequently z 2 D(A + B). Since z is an arbitrary member of D(T), we obtainD(T) ½ D(A + B). On the other hand, T , as a generator, is a closed extension ofA + B, and thus A + B ½ T .

From this theorem we see that the extension T of A+B generates a substochasticsemigroup. In particular, this is a semigroup of contractions and, consequently, T

is a dissipative operator. However, any restriction of a dissipative operator is alsodissipative, and thus A + B is dissipative. This allows a simple corollary.

Corollary 2.1. Under the assumptions of Theorem 2.1, we have:

1. T = A + B if and only if the range of ¸ ¡ (A + B); Rg(¸ ¡ (A + B)); isdense in X for some (all) ¸ > 0.

2. If additionally A + B is a closed operator; then T = A + B.

Proof. 1) Let us assume that Rg(¸ ¡ (A + B)) is dense in X. By [5, Theorem3.15], (Lumer–Phillips Theorem), this statement is equivalent to saying that A + B

is a generator (that A + B is densely defined follows from the fact that A itselfis a generator). Since T a closed extension of A + B (as a generator), we musthave A + B ½ T . Since both are generators, they must coincide. The converse isobvious.

Item 2 follows immediately.

3. SUFFICIENT CONDITION FOR T = A + B

In this section we shall sketch an approach of [2] (with some modificationdue to the author, first given for a specific situation in [4]) which enables a bettercharacterization of the generator T .

As the first step we extend all the discussed operators in the following way.


Let E denote the set of all the extended real-valued measurable functions definedon ; clearly X ½ E. We define the subset F ½ E by the following condition:f 2 F if and only if for every nonnegative and nondecreasing sequence of functions(fn)n2N satisfying supn fn = jfj we have supn(I ¡ A)¡ 1fn 2 X.

Before proceeding any further we adopt the assumption that f 2 D(B) if andonly if f+ = maxff;0g, f¡ = maxf¡ f;0g both belong to D(B). Through B weconstruct another subset of E, say G, defined as the set of all functions f 2 X suchthat for any nonnegative, nondecreasing sequence (fn)n2N of elements of D(B)such that supn fn = jfj, we have supn Bfn < +1 almost everywhere. It is easyto check that D(A) µ G µ X µ F µ E.

It can be proved that we can correctly define the mapping L : F+ ! X+ by

Lf = supn

(I ¡ A)¡ 1fn;

where 0 · fn · fn+1 for any n, and supn fn = f .To proceed, in [2] the assumptions (1)–(3) of Section 2 were supplemented by

two more:

(4) The mapping L is injective.

(5) If f 2 G+ and (f 0n)n2N, (f

00n)n2N are two nondecreasing sequences of elements

of D(A)+ satisfying f = supn f0n = supn f

00n , then supn Bf 0

n = supn Bf00n.

The common value will be denoted by Bf .

We extend the mappings L and B onto F and G, respectively, by linearity.The above construction of L is not always easy to apply. We shall interpret

it from the point of view of Sobolev towers (see, e.g., [5, Section II.5]). Thisinterpretation will also show that the assumption (4) above is superfluous.

Thus, according to [5], we define the space X¡ 1 as the completion of X withrespect to the norm

kfk¡ 1 = k(I ¡ A)¡ 1fkX:

Then the semigroup (GA(t))t¸ 0 extends continuously to a semigroup (GA;¡ 1(t))t¸ 0

in X¡ 1, which is generated by the closure of A in X¡ 1. This closure, denoted byA¡ 1 , is defined on the domain D(A¡ 1) = X ½ X¡ 1. The resolvent extends thenby density to a bounded one-to-one operator R(¸;A¡ 1) : X¡ 1 ! X¡ 1 with therange exactly equal to X . We have the following lemma.

Lemma 3.1. The operator L is a restriction of R(1; A¡ 1). As a consequence;the assumption (4) is satisfied.

Proof. Let g 2 X+ satisfy g = Lf . This means that g = supn(I ¡ A)¡ 1fn for anondecreasing sequence of nonnegative functions fn 2 X+ such that supn fn = f.

178 J. Banasiak

Since (I ¡ A)¡ 1 is a positive operator, the sequence ((I ¡ A)¡ 1fn)1n=0 is alsonondecreasing and g ¸ (I ¡ A)¡ 1fn for any n 2 N. Because g is integrable, weobtain

limn!1

Z

(I ¡ A)¡ 1fnd¹ =

Z

gd¹

and

limn!1

Z

jg ¡ (I ¡ A)¡ 1fnj =

Z

gd¹ ¡ limn!1

Z

(I ¡ A)¡ 1fnd¹ = 0:

This shows that ((I ¡ A)¡ 1fn)n2N converges in X and therefore g = Lf =R(1;A¡ 1)f . The extension for arbitrary f is done by linearity.

Corollary 3.1. If Af = ¡ mf; where m is a nonnegative; measurable; andalmost everywhere finite function; then

F = X¡ 1 = ff 2 E; (1 + m)¡ 1f 2 Xg;(3.1)

and Lf = (1 + m)¡ 1f.

Proof. Since (I ¡ A)¡ 1f = (1+ m)¡ 1f , by the definition of F, f 2 F providedsupn(1 + m)¡ 1fn 2 X for any nondecreasing sequence of nonnegative functionsfn such that supn fn = jf j.

For h 2 D(T ), let us denote g = (I ¡ T )h 2 X. Using (2.8) and replacing theoperators involved there by their extensions, we see that h can be expressed by

h =1X

k=0

L(BL)kg:(3.2)

Let us denote, for any g 2 X,

fn =nX

k=0

(BL)kg

andhn = Lfn:

We note that for a nonnegative g we can consider limits of both sequences inthe sense of monotonic convergence almost everywhere, as L and B are positiveoperators. We denote by f and h the respective limits provided they exist. In thiscase we obtain Lf = h. The result of [2] which we shall apply here reads asfollows.


Theorem 3.1. Let all the above assumptions be satisfied. If for any g 2 X+

the limit of the following real convergent sequence satisfies

limn!1

Z

A0hnd¹ ·Z

(h ¡ f +Bh)d¹;(3.3)

then T = A + B.

This theorem is not immediately useful as the function h is not explicitly known.However, the following corollary is useful in many instances.

Corollary 3.2. If Af = A1f = ¡ mf for some measurable; nonnegative; andalmost everywhere finite function m; and for any H 2 X+ such that ¡ mH +BH 2X we have

Z

(¡ mH + BH)d¹ ¸ 0;(3.4)

then T = A + B.

Proof. Inequality (3.4) coincides with (3.3) re-written in the context of this corol-lary. Hence if we prove then that any h defined by (3.2) satisfies the assumptionsimposed on H, then (3.4) will yield (3.3). Clearly, h 2 X+. Next, since fn 2 X , wehave hn 2 D(A) ½ D(B) and Bhn = fn+1 ¡ g. Thus supn Bhn = f ¡ g. On theother hand, we have f = supn fn and h = supn hn = supn(I¡ A)¡ 1fn = Lf 2 X.Thus f 2 F = X¡ 1 and from the assumption on m it follows that f is boundedalmost everywhere. This implies that h 2 G and Bh = f ¡ g. Therefore,¡ mh + Bh = h ¡ f + f ¡ g = h ¡ g 2 X. Hence, if (3.4) is satisfied forany H stipulated in the corollary, then (3.3) is satisfied for any h defined by (3.2)and the proof is complete.

Remark 3.1. The theory sketched in the previous two sections has been success-fully applied to a number of problems ranging from the linear Boltzmann equationwith external field in the context of both standard [1, 2] and extended [3] kinetictheory, Kolmogoroff’s and cell growth equations [2] and kinetic-diffusion equationsarising in the theory of hydrodynamic limits in the extended kinetic theory [4]. Inthe next section we shall apply it to the fragmentation equation.

180 J. Banasiak

4. FRAGMENTATION MODEL

An equation of a simple fragmentation model (polymer degradation) [14] canbe written as:

@tu = K®u = A® u + B® u;(4.1)

where, for x > 0,A® u = ¡ x® u(x)

and

(B® u)(x) = 2

1Z

x

y®¡ 1u(y)dy:

Here u is the number density of particles with mass x. The parameter ® is anarbitrary real number. The problem splits into three distinct cases depending onwhether ® = 0, ® < 0, ® > 0. When ® = 0, the operators K® and B® are boundedand the problem is easy; one can obtain even the closed form solution to (4.1) (cf.[7]). Therefore in the analysis below we shall always assume that ® 6= 0.

The basic space in our considerations will be the space

X1 = L1([0; 1[; xdx)

with the natural norm

kfk =

1Z

0

xjf(x)jdx:

The natural domain for K® is then

D(K® ) = D(A® ) = D® = ff 2 X1; x® f 2 X1g:

The reason for the introduction of the space X1 is that for 0 · u 2 D® ,1Z

0

(B® u)(x)xdx= 2

1Z

0

x

0@

1Z

x

u(y)y®¡ 1dy

1Adx

= 2

1Z

0

y®¡ 1u(y)

0@

yZ

0

xdx

1Ady =

1Z

0

(¡ A® u)(x)xdx:

This equation in the present context expresses the mass conservation law.We see immediately that we can apply Theorem 2.1 to get the following result.

Theorem 4.1. Let ® 2 R be arbitrary. There exists a smallest substochasticsemigroup (G®(t))t¸ 0 in X1 generated by an extension T® of K® .


Below we shall prove that the generator of (G® (t))t¸ 0 coincides with the closureof K® if ® > 0 and identify the generator (which turns out to be a proper extensionof K® ) for ® < 0.

Since the operator A® is the multiplication by ¡ x® , we can apply Corollary 3.2.Thus the space F coincides with the first associated space of the Sobolev tower, thatis, explicitly

F = X¡ 1 = L1(R+;x(1 + x®)¡ 1dx):

The set G is the set of functions f 2 X1 for which the function

F (x) = (B® f)(x) = 2

1Z

x

y®¡ 1f(y)dy

is finite almost everywhere. Clearly, the only point where F (x) could be infinite isx = 0. Now Corollary 3.2 reads that if for any 0 · H 2 X1 such that ¡ x® H +

B® H 2 X1 we have1Z

0

(¡ x® H + B®H)xdx ¸ 0;

then T® = K® . Technical details relevant to the problem at hand are given below.

Proposition 4.1. If ® > 0; then T® = K® .

Proof. Since H 2 X1 , ¡ x® H 2 L1([0; n];xdx) for any n and, since by as-sumption ¡ x® H + B® H 2 X1 , we have also B®H 2 L1([0;n];xdx). Therefore

1Z

0

(¡ x® H(x)+ (B® H)(x))xdx

= limn!1

nZ

0

(¡ x®H(x) + (B®H)(x))xdx

= limn!1

0@

nZ

0

(¡ x® H(x)xdx +2

1Z

n

0@

nZ

0

xdx

1Ay® ¡ 1H(y)dy

+2

nZ

0

0@

yZ

0

xdx

1Ay®¡ 1H(y)dy

1A

= limn!1 n2

1Z

n

y®¡ 1H(y)dy:

182 J. Banasiak

Since the limit is finite by the definition of H, it must be nonnegative and thestatement is proved.

If we look now at the case ® < 0, then a similar argument shows that1Z

0

(¡ x® H(x) + (B® H)(x))xdx

= lim²!0

1Z

²

(¡ x® H(x) + (B® H)(x))xdx

= lim²!1

0@

1Z

²

(¡ x® H(x)xdx + 2

1Z

²

0@

yZ

²

xdx

1Ay®¡ 1H(y)dy

1A

= ¡ lim²!1

²2

1Z

²

y® ¡ 1H(y)dy:

Here, unfortunately, the limit can be either zero or negative, which is inconclusive.Although it is possible to find a function H satisfying the assumptions of Corollary3.2, and for which the above limit is strictly negative (H should behave as 1 whenx is close to zero), we don’t know whether such a function belongs to D(T ).

Remark 4.1. This short section shows some advantages as well as limitations ofthe method presented earlier in this paper. Apart from the last inconclusive result,we don’t know whether Proposition 4.1 gives the best answer – in principle it isstill possible that T® = K® . Fortunately, for the model discussed in this section itis possible to find explicitly the resolvent of the generator and study it directly tofind answers to all the relevant questions. As we shall see in the next section, theresults obtained here are the best possible: for ® > 0 indeed T® = K® 6= K® ; andfor ® < 0 the generator T® is a proper extension of K® .

5. DIRECT STUDY OF THE FRAGMENTATION MMODEL

For reasons which will become clear later (see also the proof of Theorem 2.1),we shall consider a more general operator

K® ;r = A® + rB®

for 0 < r · 1. From the preceding theory, for any ® and 0 < r · 1 there is anextension T® ;r of K® ;r which is the generator of a semigroup, say, (G® ;r(t))t¸ 0 .

We shall adopt the convention that for any operator C® ;r depending on r, C® ;1 =C® .


From Section 2 it follows that for r < 1; rB® is a Miyadera perturbation of A®

and thus the Miyadera-Voigt theory implies that T® ;r = K® ;r, with the domain D® ,generating (G® ;r(t))t¸ 0.

Our next aim is to identify the resolvent of T® ;r. We start by solving the equation

¸u + x® u ¡ 2r

1Z

x

y®¡ 1u(y)dy = f:

Proceeding at this moment purely formally, we differentiate the above equation withrespect to x to get

(¸ + x® )u0 +(® + 2r)x® ¡ 1u = f 0;

and, after standard manipulations, we obtain

u =f

¸ + x® +1

(¸ + x® )1+2r=®

0@C ¡ 2r

xZ

0

f(s)(¸ + s® )¡ 1+2r=® s®¡ 1ds

1A :(5.1)

The multiplying function (¸ + x® )¡ 1 is bounded for any ® . Precisely, we have for® > 0,

(¸ + x®)¡ 1 =

½O(x¡ ® ) as x ! +1;O(1) as x ! 0+;

(5.2)

and for ® < 0

(¸ + x®)¡ 1 =

½O(x¡ ® ) as x ! 0+;O(1) as x ! +1:

(5.3)

Let us also write down the asymptotics of the kernel of the integral operator. Sincewe know that xf is integrable, we shall rather investigate (¸ + x® )¡ 1+2r=® x®¡ 2.We have for ® > 0,

(¸ + x® )¡ 1+2r=® x® ¡ 2 =

½O(x2(r¡ 1)) as x ! +1;O(x®¡ 2) as x ! 0+;

(5.4)

and for ® < 0

(¸ + x® )¡ 1+2r=® x® ¡ 2 =

½O(x®¡ 2) as x ! +1;

O(x2(r¡ 1)) as x ! 0+:(5.5)

Hence, the first term in (5.1) doesn’t create any problems. Let us consider first thecase with ® > 0. By (5.2), the multiplying function (¸ +x® )¡ 1¡ 2r=® is O(x¡ ® ¡ 2r)as x ! 1, and bounded as x ! 0+. Since u should be integrable with the weight

184 J. Banasiak

x1+® , we see that it is necessary to require that C = 21R0

f(s)(¸+s® )¡ 1+2r=® s® ¡ 1ds

and the solution takes the form

u =f

¸ + x®+

2r

(¸ + x® )1+2r=®

1Z

x

f(s)(¸ + s® )¡ 1+2r=® s®¡ 1ds:(5.6)

Let us suppose now that ® < 0. The multiplying function (¸ + x®)¡ 1¡ 2r=® isO(x¡ ®¡ 2r) as x ! 0+, and bounded away from zero as x ! +1. We have tohave u 2 X1, and thus as above the only way to ensure this is to make the bracketin (5.1) vanish at infinity, which produces the same solution given by (5.6).

Thus we introduce the operators

R® ;r(¸)f = R1(¸)f + R2(¸)f =f

¸ + x®

+2r

(¸ + x® )1+2r=®

1Z

x

f(s)(¸ + s® )¡ 1+2r=®s® ¡ 1ds:

(5.7)

In all the considerations below we assume that ¸ > 0.

Lemma 5.1. The operators R® ;r(¸) extend to bounded operators on X1.

Proof. We need to focus only on R2. Changing the order of integration, weobtain

kR2(¸)fk ·1Z

0

0@

sZ

0

x

(¸ + x® )1+2r=®dx

1A jsf(s)j(¸ + s® )¡ 1+2r=® s®¡ 2ds:

We shall analyse the behaviour of the inner integral for cases ® > 0 and ® < 0separately. Let first ® > 0. Then the inner integrand behaves as x close to zero andas x¡ ®¡ 2r+1 as x ! 1. Thus for small s the inner integral is of order of s2 , andsince by (5.4) the kernel (¸ + s® )¡ 1+2r=® s®¡ 2 behaves there as s®¡ 2, we obtainthe integrability at zero. For large s, the inner integral is of order of s¡ ® ¡ 2r+2 andsince the kernel behaves as s2r¡ 2 , we obtain that the integral is finite.

Let us look now at ® < 0. The inner integrand is of order of x¡ ®¡ 2r+1 as xis close to zero, and thus the integral is of order of s¡ ® ¡ 2r+2 . Since by (5.5) thekernel is of order of s2r¡ 2 at zero, the product is of order of s¡ ® and bounded, as® < 0; thus we obtain the integrability at zero. Next, for large x the inner integrandbehaves as x; thus the integral increases as s2 , but the kernel is of order of s®¡ 2.Hence again the product is bounded and the integral is finite.


Lemma 5.2. For each 0 < r · 1 and all ® ; the operator R® ;r(¸) is a leftinverse of ¸ ¡ K® ;r.

Proof. We have to prove that for any u 2 D® we have R® ;r(¸)(¸+A® ¡ rBa)u =u: Clearly, R1(¸)(¸ + A® )u = u. After some calculations, we obtain

¡ rR2(¸)B® u=¡ 4r2

(¸ + x® )1+2r=®

1Z

x

s® ¡ 1(¸ + s® )¡ 1+2r=®

0@

1Z

s

y®¡ 1u(y)dy

1Ads

=¡ 2r

(¸ + x® )1+2r=®

1Z

x

y® ¡ 1u(y)³(¸ + y®)2r=® ¡ (¸ + x® )2r=®

´dy

= ¡ R2(¸)(¸ + A® )u + R1(¸)(rB® )u:

Thus

R® ;r(¸)(¸ + A® ¡ rB® )u

= R1(¸)(¸ + A® )u ¡ R1(¸)(rB® )u + R2(¸)(¸ + A®)u ¡ R2(¸)(rB® )u

= u ¡ rR1(¸)B® u + R2(¸)(¸ + A® )u ¡ (¡ rR1(¸)B® u + R2(¸)(¸ + A® )u

= u

and the lemma is proved.

Proposition 5.1. For any ® 2 R and 0 < r · 1; the operator R® ;r(¸) is theresolvent of the generator of the semigroup (G® ;r(t))t¸ 0. Thus we have

T® ;r = ¸ ¡ (R® ;r(¸))¡ 1:(5.8)

Proof. For r < 1 the statement is clear as rB® is the Miyadera perturbation ofA® . In particular,

K® ;r = T® ;r = ¸ ¡ (R® ;r(¸))¡ 1:

Let now r = 1. As in the proof of Theorem 2.1 (or [13]), the semigroup (G®(t))t¸ 0

can be obtained as

G® (t)f = limr!1¡

G® ;r(t)f; f 2 X1:

Moreover, for r · r0 we have G® ;r(t) · G® ;r0 (t) for any t ¸ 0. This impliesimmediately that for f 2 X1;+,

R® ;r0(¸)f =

1Z

0

e¡ ¸tG® ;r0(t)fdt ¸1Z

0

e¡ ¸ tG® ;r(t)fdt = R® ;r(¸)f;

186 J. Banasiak

so that by (5.7) the resolvents R® ;r(¸)f monotonically converge for all f to R® (¸)f.Using again the monotonicity we obtain

kR® ;r0(¸)f ¡ R® (¸)fkX1 =

1Z

0

((R® (¸)f)(x) ¡ (R® ;r0(¸)f)(x))xdx;

and the right-hand side tends to zero by the Lebesgue monotone convergence theo-rem. Thus, for any f 2 X1;+, R® ;r(¸)f ! R® (¸)f as r ! 1¡ in X1 norm. On theother hand, it is clear that the resolvents R® ;r(¸)f converge to the resolvent R(¸)of (R® (t))t¸ 0 in X1. Thus, R(¸)f = R® (¸)f for all positive f and by linearitythis extends to the whole of X1.

Lemma 5.3. For 0 < r < 1 and all ® we have

R® ;r(¸)X1 = D® :(5.9)

If r = 1 and ® 6= 0;, then

R® (¸)X1 6= D® :(5.10)

Moreover; there exists a dense subspace X̂1 ½ X1 such that R® (¸)X̂1 ½ D® .

Proof. Let first r < 1. Then from Proposition 5.1 we see that K® ;r definedon D® is surjective. Since R® ;r(¸) is a left inverse of ¸ ¡ K® ;r, it must be theresolvent and (5.9) is proved.

Let us consider now the case r = 1 and ® > 0. Suppose 0 · f 2 X1; then weobtain as above

kx® R2(¸)fkX1 =

1Z

0

0@

sZ

0

x1+®

(¸ + x® )1+2=®dx

1A (sf(s))(¸ + s® )¡ 1+2=® s®¡ 2ds:

(5.11)

It is easy to see that for large s the inner integral behaves as ln s. As in (5.4) wesee that

(¸ + s® )¡ 1+2=® s®¡ 2 = (¸s¡ ® +1)¡ 1+2=® ! 1; as s ! 1:

Thus it is bounded away from zero for large s. Therefore, there exist functions inX1 for which the above integral is divergent (for example, one can take sf(s) =1=(s ln2 s)). Since all the functions are nonnegative, the original integral is alsodivergent by Tonelli’s theorem. It is also clear that if f has the support in [0; n],


n < 1, then the above integral is convergent. Therefore there exists a densesubspace X̂1 such that R® (¸)X̂1 ½ D® .

Let us consider now the case with r = 1 and ® < 0. Then

kx®R2(¸)fkX1 =

1Z

0

x®+1

(¸ + x® )1+2=®

1Z

x

(sf(s))(¸ + s® )¡ 1+2=® s® ¡ 2dsdx;(5.12)

and we see that the term x®+1=(¸ +x® )1+2=® behaves as x¡ 1 at zero (and as x®+1

at infinity) and therefore the above integral is divergent unless at least1R0

(sf(s))(¸ +

s® )¡ 1+2=® s®¡ 2ds = 0.

Lemma 5.4. For arbitrary ® 2 R; if for some f 2 X1 we have R® (¸)f 2 D® ;then (¸ ¡ K® )R® (¸)f = f . Therefore,

(a) if ® > 0; then the range of (¸ ¡ K® ; D® ) is dense in X1; and

(b) if ® < 0; then the range of (¸¡ K® ; D® ) is not closed; its closure is a subspaceof X1 of codimension 1.

Proof. From Lemma 5.3 we know that have R® (¸) is a left inverse to ¸ ¡ K® .Thus, if R® (¸)f 2 D® , then

R® (¸)(¸ ¡ K® )R® (¸)f = R® (¸)f:

From Proposition 5.1, we know that R® (¸) is a resolvent, and hence it is an injectiveoperator, which yields (¸ ¡ K® )R®(¸)f = f .

Let r = 1 and ® > 0. In Lemma 5.3, we showed that there exists a densesubspace X̂1 for which we have R® (¸)X̂1 ½ D(K® ). The first part of this lemmagives (¸ ¡ K® )R® (¸)X̂1 = X̂1 ½ Rg(¸ ¡ K® ) (the range of (¸ ¡ K® ; D® )), andconsequently the range of ¸ ¡ K® is dense in X1 (and not equal to X1).

Finally, let us consider the case with ® < 0 and r = 1. Suppose f 2 Rg(¸ ¡K® ), then f = (¸¡ K® )g for some g 2 D® . Lemma 5.2 implies then that R® (¸)f 2D® , but by Lemma 5.3 we must have

1R0

(sf(s))(¸ +s® )¡ 1+2=®s® ¡ 2ds = 0. In fact,

since the function (sf(s))(¸ + s® )¡ 1+2=® s®¡ 2 is integrable on [0; 1[, the function

F(x) =1Rx

(sf(s))(¸ + s® )¡ 1+2=® s®¡ 2ds is continuous and therefore it has a limit

at zero, F(0). If this limit is nonzero, then by continuity F (x) is of constant sign

188 J. Banasiak

at some interval [0; ±], and since x®+1(¸ + x® )¡ 1¡ 2=® is positive and of order ofx¡ 1 as x is close to zero, we obtain that the integral

1Z

0

x® +1

(¸ + x® )1+2=®F(x)dx(5.13)

diverges. Thus F (0) = 0 or, in other words, for f to belong to Rg(¸ ¡ K® ), it is

necessary that1R0

(sf(s))(¸ + s® )¡ 1+2=® s®¡ 2ds = 0. Hence, f must belong to the

subspace of codimension 1 which is annihilated by the functional generated by thefunction (¸ + s® )¡ 1+2=® s®¡ 1; this subspace will be denoted by ~X1. (Note that for® < 0 this function belongs to X¤

1 = ff; f measurable; x¡ 1f 2 L1([0;1[)g).However, ~X1 6= Rg(¸¡ K® ). This can be established by noting that if F (x) behaves

as 1= lnx at zero, then the integral in (5.13) diverges as0:5R0

(1=x ln x)dx = ¡ 1.

On the other hand, we can prove that ~X1 = Rg(¸ ¡ K® ). Firstly, we observe thatif f 2 ~X1 is continuous at x = 0, then f 2 Rg(¸ ¡ K® ). In fact, in this case F 0

is continuous at zero and the claim follows from±Z

0

1

xF(x)dx =

±Z

0

F (x) ¡ F (0)

xdx < +1

for sufficiently small ± > 0. Having this in mind, for a given f 2 ~X1 we constructthe sequence

fn (x) =

8>><>>:

f (x) for x > n¡ 1;

¡ 2n(¸ + x® )1¡ 2=® x¡ ® +11R

n¡ 1

f (s)(¸ + s® )¡ 1+2=® s® ¡ 1ds for (2n)¡ 1 · x · n¡ 1;

0 for 0 · x < (2n)¡ 1:

We see that 1Z

0

fn(x)(¸ + x®)¡ 1+2=® x®¡ 1dx = 0:

Moreover, since fn(x) = 0 for 0 · x · (2n)¡ 1, fn 2 Rg(¸ ¡ K® ). We shall provethat fn ! f in X1 as n ! 1. We have

kf ¡ fnkX1 ·n¡ 1Z

0

xjf(x)jdx

+2n

0B@

Z

(2n)¡ 1

n¡ 1x(¸ + x® )1¡ 2=® x¡ ® +1dx

1CA

0@

1Z

n¡ 1

jf(s)j(¸ + s® )¡ 1+2=® s®¡ 1ds

1A :


The first term tends to zero since f 2 X1. In the second term we see that theintegrand x(¸+x® )1¡ 2=® x¡ ® +1 is bounded for x close to zero, and thus the product

tends to zero due to the behaviour of1R

n¡ 1

jf(s)j(¸ + s® )¡ 1+2=® s® ¡ 1ds. Hence,

Rg(¸ ¡ K® ) = ~X1 . This ends the proof of the lemma.

Corollary 5.1. Let ® 6= 0. The operator K® is closable but not closed. If® > 0; then the extension T® of K® which generates the semigroup of (G®(t))t¸ 0

is given byT® = K® :

If ® < 0; then T® is a proper extension of K® .

Proof. It is clear that K® is densely defined. By the comment before Corollary2.1, we know that K® is dissipative for any ® . Thus from [5, Proposition II.3.14(iii-iv)] and the lemma above, we infer that K® is closable but not closed. Hence,for neither ® 6= 0, K® is the generator of a semigroup.

Assume now that ® > 0. Since we have proved that Rg(¸ ¡ K® ) is a properdense subspace of X1 , using Corollary 2.1 we obtain that T® = A® + B® is thegenerator.

Consider next the case ® < 0. Using again Proposition II.3.14 (iv) of op. cit,we see that Rg(¸ ¡ K® ) = Rg(¸ ¡ K® ). By the lemma above, Rg(¸ ¡ K® ) 6= X1.Hence in this case K® cannot be m-dissipative and therefore the generator T® mustbe a proper extension of K® .

In the last step we shall prove that the semigroup (G® (t))t¸ 0 , ® < 0, is not astochastic semigroup. This result could also be proved by modifying an argumentof [6]. but in our opinion the direct proof offered here has some instructive values.

Proposition 5.2. Let ® < 0. For any f 2 X1;+; f 6= 0; there exists t > 0 suchthat

kG® (t)fk 6= kfk:(5.14)

Therefore; (G® (t))t¸ 0 is not a stochastic semigroup.

Proof. Let f 2 X1;+, f 6= 0. If in (5.14) there was an equality for all t ¸ 0,then using the additivity of the L1 norm we would have

kR® (1)fk =

1Z

0

e¡ tkG® (t)fkdt = kfk:(5.15)

190 J. Banasiak

On the other hand,

kR® (1)fk =

1Z

0

xf(x)dx

1 + x®+2

1Z

0

0@ x

(1 + x® )1+2=®

1Z

x

f(s)(1 + s® )¡ 1+2=® s®¡ 1ds

1Adx:

(5.16)

Evaluating the second integral, we obtain

2

1Z

0

0@f(s)(1 + s® )¡ 1+2=® s®¡ 1

sZ

0

x

(1 + x® )1+2=®dx

1Ads

=

1Z

0

f(s)(1 + s® )¡ 1+2=® s®¡ 1

µs2

(1 + s® )2=®¡ 1

¶ds

=

1Z

0

f(s)(1 + s® )¡ 1s®+1ds ¡1Z

0

f(s)(1 + s® )¡ 1+2=® s® ¡ 1ds:

Inserting the above into (5.16) and simplifying, we obtain

kR® (1)fk =

1Z

0

xf(x)dx ¡1Z

0

f(x)(1 + x® )¡ 1+2=® x®¡ 1dx < kfk;

which contradicts (5.15).

REFERENCES

1. L. Arlotti, The Cauchy problem for the linear Maxwell-Boltzmann equation, J. Dif-ferential Equations 69 (1987), 166-184.

2. L. Arlotti, A perturbation theorem for positive contraction semigroups on L1-spaceswith applications to transport equations and Kolmogorov’s differential equations,Acta Appl. Math. 23 (1991), 129-144.

3. J. Banasiak, Mathematical properties of inelastic scattering models in kinetic theory,Math. Moddls Methods Appl. Sci. 10 (2000), 163-186.

4. J. Banasiak, On a diffusion-kinetic equation arising in extended kinetic theory, Math.Methods Appl. Sci. 23(14) (2000), 1237-1256.

5. K.-J. Engel, R. Nagel, One-parameter Semigroups for Linear Evolution Equations,Graduate Texts in Mathematics, Springer Verlag, New York, 1999.

6. T. Kato, On the semi-groups generated by Kolmogoroff’s differential equations, J.Math. Soc. Japan 6 (1954), 1-15.


7. W. Lamb, private communication.

8. D. J. McLaughlin, W. Lamb and A. C. McBride, A semigroup approach to fragmen-tation models, SIAM J. Math. Anal. 28 (1997), 1158-1172.

9. M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Theory: New Aspects, WorldScientific, Singapore, 1997.

10. A. Pazy, Semigroups of Linear Operators and Applications to Partial DifferentialEquations, Springer Verlag, New York, 1983.

11. H. Risken, The Fokker–Planck Equation, 2nd ed., Springer Verlag, Berlin, 1989.

12. H. H. Schaefer, Banach Lattices and Positive Operators, Springer Verlag, New York,1974.

13. J. Voigt, On substochastic C0-semigroups and their generators, Transport TheoryStatist. Phys. 16 (1987), 453-466.

14. R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation equation andpolymerization, J. Phys. A 18 (1985), 3027-3037.

School of Mathematical and Statistical Sciences,University of Natal, Durban 4041South Africa

on an extension of the kato-voigt perturbation...

Documents