The asymptotic behavior of a transport equation in cell population dynamics with a null maturation velocity

In this paper we complete the study of Rotenberg's model (M. Rotenberg, 198 3, J. Theoret. Biol. 103, 181-199) describing the growth of a cell populati on and treated partially by M. Boulanouar and H. Emamirad (Differential Int egral Equations, 13 (2000), 125-144). In contrast to our previously cited t reatment, here we impose the condition that the maturation velocity for any cell can become null. This consideration implies that the cell population never completely leaves its initial distribution, because at every time we can find some cells of initial cell population that are not divided. In thi s case, the generated semigroup is not compact. To surmount this difficulty , after studying the irreducibility of the generated semigroup, we calculat e explicitly its essential type and we show the asymptotic convergence of t he generated semigroup to a projection of rank 1. (C) 2000 Academic Press.