Analysis of a free boundary problem modeling the growth of necrotic tumors

Abstract

In this paper we make rigorous analysis to a free boundary problem modeling the growth of a necrotic tumor. A remarkable feature of this free boundary problem is that it contains two differenttype 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 tumor's 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 the analysis. 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 the outer free surface is a small perturbation of a sphere. By applying this result and some abstract theory of 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 perturbation.

Serrin’s overdetermined problem on the sphere

“…In contrast, we have to deal with it in the present paper, and it also appears in other recent works on overdetermined problems with homogeneous Neumann boundary conditions, see e.g. [13] and the references therein. Second, while (N µ ) does not possess non-constant positive solutions, most of the work on (D µ ) and (1.9) deals with positive solutions u and uses related uniqueness and nondegeneracy properties.…”

Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core

“…Asymptotic stability of stationary solutions to the above two types of necrotic tumor models was also studied, cf. [15,48]. For necrotic tumor models with (1.4), Shen et al [42] established the existence and uniqueness of radially symmetric stationary solutions to the tumor spheroid model with ν > 0 under certain conditions on the parameters.…”