Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: An application to a model of tumor growth

Abstract

We consider a free boundary problem for a coupled system consisting of an elliptic equation Ap + μ(σ -?) = 0 (p > 0) for p and a parabolic equation for σ. The problem is motivated by a model of tumor growth whereby p represents the pressure of the proliferating cells and (7 is the concentration of nutrients. On the boundary Γ(t) of the tumor region, p is equal to the surface tension, and the flux of p is equal to the normal velocity of Γ(t). In the case p = 0, the system decouples into a Hele-Shaw problem for p and a standard parabolic equation for σ. For the Hele-Shaw problem it is known that there are stationary radially symmetric solutions and each one is asymptotically stable in the following sense: If we take for initial data a small perturbation of a radially symmetric solution, then the corresponding Hele-Shaw problem has a unique global solution and its free boundary converges to a sphere as t → ∞. In this paper we prove a similar result for the coupled elliptic-parabolic problem provided p is small. The asymptotic stability result is generally false if p is not small.