Nonlinear semigroup approach to age structured proliferating cell population with inherited cycle length
Introduction
Of concern in this paper are semilinear initial-boundary value problems which model nonlinear age structured proliferating cell population dynamics of the formHere , denotes the usual norm of . represents the density at time t of the population with age a and cell cycle length l. By cell cycle length we mean the time between cell birth and cell division. The function is the rate of cell mortality. The left-hand side of the boundary condition (BC) represents the population of the daughter cells with cell cycle length l which are born from mother cells with various cycle lengths.
Lebowitz and Rubinow [9] proposed the linear equation for modelling microbial population in terms of age and cycle length formalism. In their model the cell cycle length l of individual cell is assumed to be an inherent characteristic determined at birth. Webb [13] treated this linear equation in the space of continuous functions and showed that the solution has the asynchronous exponential growth. Latrach and Mokhtar-Kharroubi [7] have dealt with the spectral analysis of a linear operator corresponding to the linear problem in . They show that the linear operator generates a semigroup in with compactness assumption on K. Boulanouar [2], [3] demonstrated that the linear operator mentioned above generates a semigroup in without restriction and studied the irreducibility and the asymptotic behavior of the semigroup. Jeribi [5] and Latrach et al. [8] studied a stationary problem related to a nonlinear version of Lebowitz–Rubinow model.
Our objective here is to present a nonlinear version of Lebowitz–Rubinow model (DE)–(BC) as abstract semilinear problemsin , , which are coupled with nonlinear constraints of the formand to construct a semigroup, denoted , of Lipschitz operators in which provides solutions in a strong or generalized sense to the evolution problem (SE)–(NC).
These constraints (NC) are formulated in a possibly different Banach space . Here A is a linear operator in X which represents a linear differential operator subject to suitable boundary conditions, and are linear operators from the domain of A into Z, F is a possibly nonlinear but continuous perturbing operator in X and K is a nonlinear continuous operator from into Z which specifies the nonlinear constraints in Z. The operators and stand for unbounded operators such as trace operators in our concrete problem, and so they need not be continuous.
Although F and K are continuous and (SE) is a typical semilinear evolution equation in X, the problem for (SE)–(NC) cannot be formulated as a standard semilinear evolution problem, since (NC) contains a composition of K and an unbounded operator . In fact, we necessitate treating the problem for (SE)–(NC) as a fully nonlinear evolution problem rather than a semilinear evolution problem. Moreover, the resulting semigroup is not quasicontractive with respect to the original norm but it is locally equi-Lipschitz continuous on X in the sense that for any and any bounded set B there exist constants and such that for , and .
The feature of our argument is to apply a generation theorem of quasicontractive nonlinear semigroup with respect to a family of equivalent norms . In the case of , the linear operator A is restricted to set of elements in for which (NC) holds and the original problem is converted to a fully nonlinear problem which we call (NP; p) in this paper. In the case of , the problem is reformulated as a formal semilinear evolution problem in the product space , although the semilinear operator is no longer a continuous perturbation of the linear unbounded operator. In both cases, appropriate equivalent norms should be employed to show the local quasidissipativity and subtangential conditions of the governing operators and then apply a generation theory for locally quasicontractive semigroups. For another approach to this type of semilinear problem we refer to [12].
Our paper is organized as follows: Section 2 outlines the main points of a generation theory for locally quasicontractive semigroups with respect to a family of equivalent norms. In Section 3, the fundamental assumptions for the models are made and a full statement of our main theorem (Theorem 3.1) is given. Basic lemmas which we need to prove our main result are discussed in Section 4. Section 5 is devoted to the proof of Theorem 3.1 in the case of . The proof in the case of is given in Section 6 by reformulating the original problem in the product space .
Section snippets
Preliminaries
In this section we outline the main points of a generation theory of nonlinear semigroups (developed in [6]) in our context. This general theory is effectively applied to the proof of our main result. For detailed arguments, we refer the readers to [10].
It should be noted here that the use of an appropriate family of equivalent norms (as introduced in (H)) suggests our specific approach to treat nonlinear semigroups such as those in the class defined later.
Let be a Banach space.
Age structured proliferating cell population dynamics
In this section we make basic assumptions for the initial-boundary value problem (DE)–(BC) and give the statement of our main result. Here we put the following assumptions on , , d, k and c.
(c.1) The minimum cell cycle length is positive and the maximum cell cycle length is finite. for and . , and for .
(c.2) For each fixed and , , and are measurable in a, l, and
Basic lemmas
In order to apply theorems stated in Section 2 to the nonlinear problem (DE)–(BC), we need the eight lemmas below: Lemma 4.1 Let . For each the traces and exist as functions in . Moreover, and are bounded linear operators from equipped with graph norm into . Proof Let be a non-increasing function satisfying for and for . Let . By using the identities
Generation of nonlinear semigroups in ,
In this section we construct nonlinear semigroups in for through Theorem 2.2. To this end, we check the assumptions of Theorem 2.2. Lemma 5.1 For and there exists a constant such that to each and there is satisfying Proof In order to prove this lemma, we consider a product Banach space with norm . Let and . Set and let be the maximum of
Generation of a nonlinear semigroup in
In this section we convert the problem (NP;1) into the Cauchy problem in the product Banach space and then show the existence of a nonlinear semigroup in which gives a unique mild solution to (NP;1).
Let be the product Banach space with norm . Put . We introduce a linear operator in by and a nonlinear operator by Then the operator is locally quasidissipative
References (13)
Etude d’un modèle de Lebowitz-Rubinow
C. R. Acad. Sci. Paris Sér. I
(1999)A mathematical study in the theory of dynamic population
J. Math. Anal. Appl.
(2001)A nonlinear problem arising in the theory of growing cell populations
Nonlinear Anal. Real World Appl.
(2002)- et al.
On an unbounded linear operator arising in the theory of growing cell population
J. Math. Anal. Appl.
(1997) Sobolev Spaces
(1975)Perturbing the boundary conditions of a generator
Houston J. Math.
(1987)
Cited by (6)
Nonlinear semigroup approach to transport equations with delayed neutrons
2018, Acta Mathematica ScientiaWell-posedness of a nonlinear model of proliferating cell populations with inherited cycle length
2016, Acta Mathematica ScientiaOn the solutions for a nonlinear boundary value problem modeling a proliferating cell population with inherited cycle length
2016, Nonlinear Analysis, Theory, Methods and ApplicationsExistence and uniqueness results for a nonlinear evolution equation arising in growing cell populations
2014, Nonlinear Analysis, Theory, Methods and ApplicationsCitation Excerpt :In Section 5 we recall the main results concerning accretive operators as well as gather some facts from the functional analysis which are required in the remainder of the paper. Mathematical models for nonlinear age structured dynamics with nonlocal boundary were already considered in [14,15]. Let us point out that the book [16] provides a detailed account for transport equations in Biology where the mathematical analysis of various kinds of such equations was derived.
An existence and uniqueness principle for a nonlinear version of the Lebowitz-Rubinow model with infinite maximum cycle length
2018, Mathematical Methods in the Applied SciencesA Nonlinear Age-Structured Model of Population Dynamics with Inherited Properties
2016, Mediterranean Journal of Mathematics
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Partially supported by a Grant-in-Aid for Scientific Research (B)(1) No. 16340042 from JSPS.