The Asymptotic Behavior of a Transport Equation in Cell Population Dynamics with a Null Maturation Velocity

• Published in: Journal of mathematical analysis and applications Vol. 243; no. 1; pp. 47 - 63
• Main Authors: Boulanouar, M., Emamirad, H.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.03.2000; Elsevier
• Subjects: Animal and plant ecology; Animal, plant and microbial ecology; Biological and medical sciences; Demecology; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects; Mathematical analysis; Mathematics; Partial differential equations; Sciences and techniques of general use; Transport equation; Lebowitz Rubinow model; Reducibility; Banach lattice; Sobolev space; Asymptotic behavior; Reproduction rule; Semigroup; Boundary condition; Banach space; Linear operator; Partial differential equation; Exponential growth; Convergence; Cauchy problem; Ripening; Cell population; Population dynamics
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1006/jmaa.1999.6650

In this paper we complete the study of Rotenberg's model (M. Rotenberg, 1983, J. Theoret. Biol.103, 181–199) describing the growth of a cell population and treated partially by M. Boulanouar and H. Emamirad (Differential Integral Equations, 13 (2000), 125–144). In contrast to our previously cited treatment, 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 this case, the generated semigroup is not compact. To surmount this difficulty, after studying the irreducibility of the generated semigroup, we calculate explicitly its essential type and we show the asymptotic convergence of the generated semigroup to a projection of rank 1.