  In this paper we study asymptotic behavior of solutions for a
multidimensional free boundary problem modelling the growth of nonnecrotic
tumors. We first establish a general result for differential equations in
Banach spaces possessing a local 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
$D_A(\theta,\infty)$ 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 $\gamma^\ast$ then the unique stationary solution is
asymptotically stable modulo translations, provided the constant $c$
representing the ratio between the nutrient diffusion time and the tumor-cell
doubling time is sufficiently small, whereas if $\gamma< \gamma^\ast$ then this
stationary solution is unstable.
