Journal of Mathematical Analysis and Applications 235, 435᎐453 Ž1999. Article ID jmaa.1999.6365, available online at http:rrwww.idealibrary.com on
An Abstract Neutral Functional Differential Equation Arising From a Cell Population Model O. Arino Laboratoire de Mathematiques Appliquees, ´ ´ I.P.R.A., A¨ . de l’Uni¨ ersite, ´ 64000 Pau, France
and O. Sidki Faculte´ des Sciences et Techniques Fes-Saıss, ¨ B.P. 2202, Fes, Maroc Submitted by G. F. Webb Received July 9, 1998
In this article, an abstract Žinfinite dimensional. neutral functional differential equation arising from a cell population model is exhibited. It is then shown that a large class of such equations can be solved by means of the theory of nonlinear semigroups. Finally, application to the model equation is detailed. 䊚 1999 Academic Press
1. INTRODUCTION In this article, we consider the model of cell proliferation described by the equation nŽ t , x . s 2 H N Ž t .
q⬁
q⬁
H0 H0
f Ž x, Ž , . . ␥ Ž , . n Ž t y , . d d ,
Ž 1. in which NŽ t. s
q⬁
q⬁
t
H0 H0 Htyn Ž s, y . ␥ Ž , y . ds d dy. 435 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
436
ARINO AND SIDKI
This model is a modification of a model first proposed in Kimmel et al. w9x. The original model was linear. Nonlinear variants were later considered in Arino and Kimmel w1, 2x and Arino and Mortabit w4x. The model is based on the subdivision of the cell cycle into four consecutive phases represented as follows: ª y G1
S
G2
; ␥ Ž ., y .
M ª y ; f Ž ⭈, x . ª x s ⌽Ž , y. .
During its progression inside the cycle, a cell keeps growing, although with a variable strength. Generally, it doubles its size from birth Žas a daughter cell. to the end of the G2 phase before the M phase Žmitosis. were it divides into two identical cells, each with half the constituents of the mother cell. The main hypothesis introduced in w9x, in the case of subdivision of the cell, is that the division is not equal. Equation Ž1. takes into consideration two further aspects, compared to the one aspect in w9x. One assumes here that the life duration of a cell Žthat is, the length of a cycle. is not determined by the initial size, but dependence is, in probability, determined by a conditional density ␥ Ž., . Žconditioned on the size.: H12␥ Ž , . d s Probability for the lifelength of a cell, with initial size , to lie within the interval w 1 , 2 x. As a result, the final size of cells cannot be expressed in terms of their initial size only Žas was assumed in w9x. but it is also a function of the lifelength: Ž , .. A linear model based on these considerations was presented in w3x. Here, we introduce the limiting effects due to the environment in the form of a function of the total population, which decays to zero when population grows to q⬁. Such models were considered in w1, 2x. Equation Ž1. collects and extends two previous models: a linear by Arino, et al. w3x and a nonlinear by Arino and Kimmel w1x. Equation Ž1. is an integral equation and was studied as such in w1, 3x. Here we transform the integral equation into a functional differential equation of neutral type ŽNFDE. that we solve under suitable hypotheses. Thus, we obtain both a strict extension of previous existence results w1, 3x and novel regularity properties verified by the solutions of the NFDE. Differentiating Žformally. Eq. Ž1. with respect to time yields the NFDE,
⭸ ⭸t
n Ž t , .. s G Ž n t . Ž .. ,
Ž 2.
437
NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION
where G is a nonlinear operator defined from W 1, 1 Žwyr, 0x; L1 Ž0, q⬁.. into L1 Ž0, q⬁. by G Ž . s 2 K Ž . H˙ L Ž . =
q⬁
q⬁
f Ž ., Ž , . . ␥ Ž , . Ž y , . d d
H0 H0
q 2H L Ž. =
q⬁
q⬁
f Ž ., Ž , . . ␥ Ž , .
H0 H0
with
¡K Ž . s H H H ~ ¢L Ž . s H H H q⬁
0
q⬁
0
q⬁
0
q⬁
0
0
⭸
y
⭸
0
y
⭸ ⭸
Ž y , . d d Ž 3 .
Ž , . ␥ Ž , y . d d dy ,
Ž , . ␥ Ž , y . d d dy
for every g W 1, 1 Žwyr, 0; L1 Ž0, q⬁... Under appropriate assumptions on the parameters defining Eq. Ž1., one shows that, for each initial value n 0 g W 1, 1 Žwyr, 0x; L1 Ž0, q⬁.. and each T ) 0, NFDE Ž2. possesses one and only one solution n, n g W 1, 1 Žwyr, T x; L1 Ž0, q⬁... Integrating Ž2., one obtains the solution of Ž1. and, in the same way as in w2x, one shows that it is nonnegative for all t G 0, if n 0 G 0. 2. RESOLUTION OF THE NFDE: ˙ xŽ t . s FŽ x t . In order to study Eq. Ž2., we give some results on the following class of NFDEs. dx dt
s F Ž xt . ,
x 0 s g W 1, 1 Ž w yr , 0 x ; X . ,
0 F t F T,
Ž 4.
where x: wyr, T x ª X, 0 - r - q⬁ is the delay, X is a Banach space with norm
