Existence and Asymptotic Behaviour of Solutions of Differential and Integral Equations in Some Function SpacesView this Special Issue
Spectral Distribution of Transport Operator Arising in Growing Cell Populations
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 : 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., ) 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.  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  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  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 ). 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 ). 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 One. 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 .
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 ; 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