Asymptotic Stability of Stationary Solutions of a Free Boundary Problem Modeling the Growth of Tumors with Fluid Tissues

Abstract

This paper aims at proving asymptotic stability of the radial stationary solution of a free boundary problem modeling the growth of nonnecrotic tumors with fluid-like tissues. In a previous paper we considered the case where the nutrient concentration $σ$ satisfies the stationary diffusion equation $Δσ=f(σ)$, and proved that there exists a threshold value $γ_*>0$ for the surface tension coefficient $γ$, such that the radial stationary solution is asymptotically stable in case $γ>γ_*$, while unstable in case $γ<γ_*$. In this paper we extend this result to the case where $σ$ satisfies the non-stationary diffusion equation $\epsln\partial_tσ=Δσ-f(σ)$. We prove that for the same threshold value $γ_*$ as above, for every $γ>γ_*$ there is a corresponding constant $\epsln_0(γ)>0$ such that for any $0<\epsln<\epsln_0(γ)$ the radial stationary solution is asymptotically stable with respect to small enough non-radial perturbations, while for $0<γ<γ_*$ and $\epsln$ sufficiently small it is unstable under non-radial perturbations.