Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either D-A(theta)or D-A(theta, infinity) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient gamma is larger than a threshold value gamma* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if gamma < gamma* then this stationary solution is unstable. (c) 2008 Elsevier Inc. All rights reserved.