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If a differential equation in a Banach manifold is invariant orquasi-invariant under the action of one or more Lie groups, then its stationarypoints cannot be isolated, so that classical linearized stability theorem doesnot apply to it. The first main purpose of this paper is to establish alinearized stability theorem for parabolic differential equations in Banachmanifolds which are either invariant or quasi-invariant under actions of anumber of Lie groups. The second purpose of this paper is to apply this theoremto analyze stability of stationary solutions of some free boundary problems. Inorder to apply the abstract result to concrete free boundary problems, Banachmanifold made up of certain kind of domains such as simple domains in${\mathbf{R}}^n$ is a fundamental tool which seems to have not beenwell-studied in the literature yet. Hence in this paper we also make some basicinvestigation to a such manifold. In Section 5 we use Nash-Moser implicitfunction theorem to prove an interesting result for an obstacle problem whichsays that if the domain $\Omega$ of this obstacle problem is a smallperturbation of a sphere then its interface $\Gamma$ is smooth and depends on$\Omega$ smoothly. By using these results, in the last section we proveasymptotic stability of radial stationary solution of a free boundary problemmodeling the growth of necrotic tumors, which has been kept open for over tenyears.

Linearized stability theorem for invariant and quasi-invariant parabolic  differential equations in Banach manifolds with applications to free boundary  problems