Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate mu and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius R-S which is independent of it. It was recently proved that there is a function mu(*)(RS) such that the spherical stationary solution is linearly stable if mu < mu(*)(RS) and linearly unstable if mu > mu(*)(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if mu < mu(*)(RS). (c) 2005 Elsevier Inc. All rights reserved.