  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.
