Analysis of a free boundary problem modeling the growth of necrotic tumors (1902.04066v2)

Abstract: In this paper we make rigorous mathematical analysis to a free boundary problem modeling the growth of necrotic tumors. A remarkable feature of this free boundary problem is that it contains two different-type free surfaces: One is the tumor surface whose evolution is governed by an evolution equation and the other is the interface between the living shell of the tumor and the necrotic core which is an obstacle-type free surface, i.e., its evolution is not governed by an evolution equation but instead is determined by some stationary-type equation. In mathematics, the inner free surface is induced by discontinuity of the nonlinear reaction functions in this model, which causes the main difficulty of analysis of this free boundary problem. Previous work on this model studies spherically symmetric situation which is in essence an one-dimension free boundary problem. The purpose of this paper is to make rigorous analysis in general spherically asymmetric situation. By applying the Nash-Moser implicit function theorem, we prove that the inner free surface is smooth and depends on the outer free surface smoothly when it is a small perturbation of the surface of a sphere. By applying this result and some abstract results for parabolic differential equations in Banach manifolds we prove that the unique radial stationary solution of this free boundary problem is asymptotically stable under small non-radial perturbations.