Lie Group Action and Stability Analysis of Stationary Solutions for a Free Boundary Problem Modelling Tumor Growth

Abstract

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.