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 Correspondence should be addressed to Hongxing Wu; jxsruwhx@163.com 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. Transport equation with partly smooth boundary conditions arising in growing cell populations is studied in Lp (1 < p < +∞) space. It is to prove that the transport operator AH generates a C0 semigroup and the ninth-order remainder term R9(t) of the Dyson-Phillips expansion of the semigroup is compact, and the spectrum of transport operator AH consists of only finite isolated eigenvalues with finite algebraic multiplicities in a trip Γω. The main methods rely on theory of linear operators, comparison operators, and resolvent operators approach.


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  (, V, 0) =  0 (, V) ,  0 (0, V, ) =   (, V, ) , where  is maturity degree of cells,  ∈ [0,],  > 0; the degree of maturation  is then defined in the manner that  = 0 at the birth and  =  at the death, and their maturation velocity V, V  ∈ [0, ],  > 0, (, V, ) describes the number density of cell population as a function of the degree of maturation ; the maturation velocity V and the time , (, V) denote the total transition cross-section while the function (, V, V  ) represents the transition rate at which cells change their velocities from V to V  , (, V, 0) =  0 (, V) is initial condition,  is linear operator in boundary space and is known as transition rule in biology.
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][11][12][13][14][15][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  1 spaces and proved that the remainder term was compact and got Wang and Cheng [13] whose norm is We denote by  0  ,    the following boundary spaces: where In the sequel,  0  and    will often be identified with   ([0, ]; VV).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  0 be the real defined by Consider now the resolvent equation for operator   : where  is a given function of   .For Re  > − 0 the solution is formally given by Accordingly, for  = , we get Now, let us define the transport operators   ,   ,   , and   by (20) Because of (18), we can write Let  0 be a real number defined by Clearly, for Re  > − 0 , we get Because of ( 17) and (23) we get Accordingly, for Re  > − 0 , the resolvent of the operator   is given by In the following, we assume that  and  satisfy the following: (O 1 )  =  1 +  2 , where  1 is bounded and positive operator and  2 is compact and positive operator; (O 2 )  is regular operator in   .So it can be approximated in the uniform operator topology by operators; then where   (⋅) ∈  ∞ ([0, ], ),   (⋅) ∈   ([0, ], V),   (⋅) ∈   ([0, ], V), (1/) + (1/) = 1, and  is finite set.
Indeed, let us first observe that if we replace in the definition of   the function (, V) by the real  0 , we obtain a new streaming operator which we denote by   .Arguing as above we can define the operators   ,   ,   , and   , which satisfy, for any Re  >  0 , Lemma 1.For any  > 0,  ∈ N, then where   () and   () denote the nth remainder term of the Dyson-Phillips expansion of the semigroup generated by   and   .
Proof.For all Re  >  0 , a simple calculation shows that For ( 25), ( 27), and (29) we get Let  > 0 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   () ≥0 and   () ≥0 denote the strongly continuous semigroup generated by   and   .The positivity of  and (30) imply that Now, we define transport operator   by Because The exponential formula for strongly continuous semigroups leads to where   () ≥0 and   () ≥0 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: is compact for all  such that Re  ≥ ; (2) for every Then  2+1 () is compact on  for each  > 0.

Main Result
In this section we are ready to prove the main result of this paper.Let uniformly on Γ  .
Proof.Now we are going to divide the proof into several steps.
Step Three.For any  ∈ N, for  ∈ [0, 1), prove that uniformly on Γ  .For all  ∈   , easy calculations show that Let where Because  1 and  2 are bounded operators, it suffices to prove that uniformly on Γ  .In the same principle as (75), we can get (87) immediately.
Step Four.For any Re  >  0 , because of (30) and assumption O 2 , we have According to (25)-( 27)-( 29)-( 30)-(43), it suffices to establish that, for  ∈ [0, 1), prove that lim uniformly on Γ  .Note that    1 and    2 do not commute, so we have Let Since ( −   ) −1 is bounded operator and we get that Π() is compact operator on   .On the other hand, because of thanks to (29), and for Re ≥  0 + , we have uniformly on Γ  .Now, applying Lemma 3 we conclude that, for each  > 0,  9 () 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  >  0 , then the operator ( −    1 ) −1 is bounded invertible and (104) becomes where Step Two.Prove that ( − Ξ  ) −1 exists for  in the half plane with | Im | sufficiently large.
(i) Let  be an element of and set We consider  ∈    ; then (112) show that uniformly on Γ 0 .This ends Step two by (iii).