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In this paper we make rigorous mathematical analysis to a free boundaryproblem modeling the growth of necrotic tumors. A remarkable feature of thisfree boundary problem is that it contains two different-type free surfaces: Oneis the tumor surface whose evolution is governed by an evolution equation andthe other is the interface between the living shell of the tumor and thenecrotic core which is an obstacle-type free surface, i.e., its evolution isnot governed by an evolution equation but instead is determined by somestationary-type equation. In mathematics, the inner free surface is induced bydiscontinuity of the nonlinear reaction functions in this model, which causesthe main difficulty of analysis of this free boundary problem. Previous work onthis model studies spherically symmetric situation which is in essence anone-dimension free boundary problem. The purpose of this paper is to makerigorous analysis in general spherically asymmetric situation. By applying theNash-Moser implicit function theorem, we prove that the inner free surface issmooth and depends on the outer free surface smoothly when it is a smallperturbation of the surface of a sphere. By applying this result and someabstract results for parabolic differential equations in Banach manifolds weprove that the unique radial stationary solution of this free boundary problemis asymptotically stable under small non-radial perturbations.

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