Journal of Function Spaces

Volume 2014 (2014), Article ID 748792, 10 pages

http://dx.doi.org/10.1155/2014/748792

Research Article

Spectral Distribution of Transport Operator Arising in Growing Cell Populations

Hongxing Wu, Shenghua Wang, and Dengbin Yuan

Department of Mathematics, Shangrao Normal University, Shangrao, Jiangxi 334001, China

Received 27 May 2014; Accepted 12 August 2014; Published 25 August 2014

Academic Editor: Leszek Olszowy

Copyright © 2014 Hongxing Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Transport equation with partly smooth boundary conditions arising in growing cell populations is studied in space. It is to prove that the transport operator generates a semigroup and the ninth-order remainder term of the Dyson-Phillips expansion of the semigroup is compact, and the spectrum of transport operator consists of only finite isolated eigenvalues with finite algebraic multiplicities in a trip . The main methods rely on theory of linear operators, comparison operators, and resolvent operators approach.

1. Related Knowledge

In this paper, we are concerned with the following transport equation, which was proposed by Rotenberg in [1]:

with the initial condition and the general biological rule

where is maturity degree of cells, , ; the degree of maturation is then defined in the manner that at the birth and at the death, and their maturation velocity , , describes the number density of cell population as a function of the degree of maturation ; the maturation velocity and the time , denote the total transition cross-section while the function represents the transition rate at which cells change their velocities from to , is initial condition, is linear operator in boundary space and is known as transition rule in biology.

It is well known that the streaming operator generates a strongly continuous semigroup () (see, e.g., [2–5]). If the collision operator is bounded, then the classical perturbation theory (see, [6–8]) shows that the transport operator generates also a strongly continuous semigroup () (see, e.g., [9]) given by the Dyson-Phillips expansion:

where and

where represents disturbance operators, and the remainder term is given by

As is well known, the solution of (1)-(2) exists and is unique for all ; here represents transport operator with boundary condition .

For the past few years, there are many research works on the spectrum analysis of (1), and some of them had been discussed in [6, 10–16]. Jeribi et al. [10] discussed Rotenberg’s model of cell population with general compact boundary conditions and proved that the transport semigroup was irreducible, and a spectral decomposition of the solution into an asymptotic term was derived. Latrach and Megdiche [11] discussed the large time behavior of the solution to the Cauchy problem governed by a transport equation with Maxwell compact boundary conditions arising in growing cell population in spaces and proved that the remainder term was compact and got

Wang and Cheng [13] had proved that the transport operator with compact integral boundary conditions generated a semigroup and the existence of the eigenvalues in    space in L-R’s model. So, it is natural to set the following question: what happens about the transport operator spectral distribution in Rotenberg’s model when boundary conditions are partly compact? In this paper, we discuss the transport equation with partly smooth boundary conditions arising in growing cell populations in    space. It is to prove that the transport operator generates a semigroup and the ninth-order remainder term of the Dyson-Phillips expansion of the semigroup is compact, and the spectrum of transport operator consists of only finite isolated eigenvalues with finite algebraic multiplicities in a trip .

Now, we introduce different notions and notations. Let us first make precisely the functional setting of the problem. Let

whose norm is

We denote by the following boundary spaces:

where

In the sequel, and will often be identified with . We define the partial Sobolev space as follows:

We define the disturbance operators and streaming operator by

So we may define the transport operator by

Let be the real defined by

Consider now the resolvent equation for operator :

where is a given function of . For the solution is formally given by

Accordingly, for , we get

Now, let us define the transport operators , , , and by

Clearly, for satisfying , the operators , , , and are bounded and positive, and it is not difficult to check that

Because of (18), we can write

Let be a real number defined by

Clearly, for , we get

Because of (17) and (23) we get

Accordingly, for , the resolvent of the operator is given by

In the following, we assume that and satisfy the following:

(O₁), where is bounded and positive operator and is compact and positive operator;(O₂) is regular operator in . So it can be approximated in the uniform operator topology by operators; then

 where , , , , and is finite set.

Indeed, let us first observe that if we replace in the definition of the function by the real , we obtain a new streaming operator which we denote by . Arguing as above we can define the operators , , , and , which satisfy, for any ,

Lemma 1. For any , , then

where and denote the nth remainder term of the Dyson-Phillips expansion of the semigroup generated by and .

Proof. For all , a simple calculation shows that

For (25), (27), and (29) we get

Let be a fixed real; by (30), it is obvious that, for all integer and all such that , we have

Consequently,

Next, in view of the exponential formula for strongly continuous semigroups, we have

where and denote the strongly continuous semigroup generated by and . The positivity of and (30) imply that

Now, we define transport operator by

Because

therefore

The exponential formula for strongly continuous semigroups leads to

where and denote the strongly continuous semigroup generated by and . Because of (12),

As a consequence of (12)–(33)–(38)-(39), we have (28) immediately.

Lemma 2 (see [9]). Let B be the generator of a strongly continuous semigroup on a Banach space , and denote the bounded linear operators in . Assume that there exist and satisfying the following:

(1) is compact for all such that ;(2)for every satisfying , then

 Then is compact on for each .

Lemma 3 (see [12]). Let be a positive measure space and let be bounded linear operators on . If is compact and , then is compact.

2. Main Result

In this section we are ready to prove the main result of this paper. Let

Theorem 4. If assumptions O₁ and O₂ are satisfied, for , then

uniformly on .

Proof. Now we are going to divide the proof into several steps.

Step O ne. For any , prove that

uniformly on . For all ,

Using the change of variable , for all , we have

Now, we define a bounded linear operator by

For all , clearly, converges in the operator topology to uniformly on as . So, it suffices to show that, for ,

uniformly on . Putting

we get

Let , where

Since and are bounded operators, it suffices to prove that

uniformly on . For all ,

Applying the Hölder inequality we get

Then we have

Putting , , and , we get

so we have

For any ,

and we get

So, it remains to show that

uniformly on . We define now the functions

such that , , , and . So, it suffices to establish that

uniformly on . Because the functions and are simple functions, we fix , and for all , we have

where , , is nonnegative constant and . We get

where

This implies that

Consequently, we have

where

Finally, we get

So we can get (43) immediately.

Step Two. For any , for , prove that

uniformly on . Since is positive and compact operator, so it suffices to establish the result for a positive operator of rank one; that is,

where   ,   , and . So for any , , we get

Let

where

Since and are bounded operators,

so it suffices to prove that

uniformly on . For any , , let

we defined a bounded linear operator by

Let and ; then

where and

So, we can assume that

where . Clearly, using the Lebesgue dominated convergence theorem, we get the nets of operators   . And for all , it suffices to prove that

uniformly on . Because

we get

And we get

Applying the Hölder inequality we get

And we have

So we can get (69) immediately.

Step Three. For any , for , prove that

uniformly on . For all , easy calculations show that

Let

where

Because and are bounded operators, it suffices to prove that

uniformly on . In the same principle as (75), we can get (87) immediately.

Step Four. For any , because of (30) and assumption O₂, we have

According to (25)–(27)–(29)-(30)–(43), it suffices to establish that, for , prove that

uniformly on . Note that and do not commute, so we have

where each is the product of n factors involving both and except the term

So, for , the operator appears at least one time in the expression of . So, there exists , such that

where is uniformly bounded on . Now using the inequality

and (53), for , , we get

uniformly on . On the other hand, for , we have

Using (69) again, it remains to prove that

uniformly on . This follows from (87). This ends the proof.

Theorem 5. If assumptions O₁ and O₂ are satisfied and , then the spectrum of transport operator consists of only finite isolated eigenvalues which have finite algebraic multiplicities in trip ; furthermore a real exists such that , where denotes point spectrum set of transport operator .

Proof. Consider the following.

Step One. Prove that the spectrum of the transport operator consists of, at most, finite isolated eigenvalues which have finite algebraic multiplicities in trip . By Lemmas 1–3, it suffices to prove that is compact operator on .

On one hand, because of hypothesis , we have and is compact operator on . Let

Since is bounded operator and

we get that is compact operator on .

On the other hand, because of

thanks to (29), and for , we have

So for , together with (102), (103), and (104) we have

By Theorem 4, we get

uniformly on . Now, applying Lemma 3 we conclude that, for each , is compact operator on , together with [3]; we end Step one. Now we consider again the resolvent equation which is equivalent to solving in the following one:

If , then the operator is bounded invertible and (104) becomes

where

Step Two. Prove that exists for in the half plane

with sufficiently large.

Let be an element of

and set

where and are the dual operators of and . Easy calculations show that the dual of the operator is . Let

be a sequence in such that

We consider ; then (112) show that

By the Riemann-Lebesgue theorem, we obtain

a.e. on . Furthermore, for every integer , we have

By the dominated convergence theorem, we get

for all   .

Prove that the family , , is collectively compact. Let be the unit ball of the space and let be a sequence in ; then there exists a sequence in such that

Since is compact and is bounded in the space , there exists a converging subsequence of in . This is the wanted.

The use of and together with Anselone’s Proposition [17, Proposition 3.1] gives

uniformly on . Then

uniformly on . Since

therefore, we obtain

uniformly on . This ends Step two by .

Step Three. We will prove that .

First, because of

there exists for and we get

Hence

Because is compact on and

we get that is compact on , and we have

And we get

Next, by spectral mapping theorem, we get that is continuous decreasing function on , since

Because is continuous decreasing function on , it suffices to prove that

Let

we have   , since

and we get

Due to the hypothesis , we get

Finally, according to (135) and the fact that is continuous decreasing function on , we know that there must be a real such that

Hence,   . This ends the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their gratitude to the referee for his/her valuable suggestions. The research was supported by National Natural Science Foundation of China (no. 11461055), the Natural Science Foundation of Jiangxi Province of China (no. 20132BAB201002) and the Office of Education of Jiangxi Province of China (no. GJJ13706).

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