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 $\sigma$ satisfies the stationary diffusion equation $\Delta \sigma =f(\sigma )$, and proved that there exists a threshold value ${\gamma }_{\ast }>0$ for the surface tension coefficient $\gamma$, such that the radial stationary solution is asymptotically stable in case $\gamma >{\gamma }_{\ast }$, while unstable in case $\gamma <{\gamma }_{\ast }$. In this paper we extend this result to the case where $\sigma$ satisfies the non-stationary diffusion equation $\epsln\partial_tσ=Δσ-f(σ)$. We prove that for the same threshold value ${\gamma }_{\ast }$ as above, for every $\gamma >{\gamma }_{\ast }$ 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 $0<\gamma <{\gamma }_{\ast }$ and $\epsln$ sufficiently small it is unstable under non-radial perturbations.