Compactness Properties and Asymptotics of Strongly Coupled Systems

Abstract

Recently Gyllenberg and Webb investigated a model which is designed to describe the growth of a solid tumor. The main biological observation used in their model is the fact that a tumor consists of two functionally different kinds of cells: (a) growing and proliferating cells, and (b) resting cells forming the necrotic kernel of the tumor. The linear version of the model can be formulated as a system of differential equations. Gyllenberg and Webb studied the asymptotic behavior of a tumor and proved that their model leads-using rather weak assumptions-to "balanced"or "asynchronous exponential growth" in a strong sense. Motivated by this example we develop some new functional analytic tools in order to prove "balanced exponential growth" of systems of partial differential equations. To this purpose we describe systems by using operator matrices [formula] on a product space X = E × F, where E and F are Banach lattices. To determine the asymptotic behavior of the corresponding operator semigroup (etK)t≥0 and to prove "balanced exponential growth" we introduce conditions for the matrix entries A, B, C, D which lead to a certain compactness property - essential compactness - of the solution semigroup. Together with some further knowledge about the spectrum of K and the resolvent (λI − K)−1 we can conclude "balanced exponential growth" of the semigroup. Finally these results are applied to the above model for the growth of a tumor cell population.