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

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 ∆σ = 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 ε∂tσ = ∆σ−f(σ). We prove that for the same threshold value γ∗ as above, for every γ > γ∗ there is a corresponding constant ε0(γ) > 0 such that for any 0 < ε < ε0(γ) the radial stationary solution is asymptotically stable with respect to small enough non-radial perturbations, while for 0 < γ < γ∗ and ε sufficiently small it is unstable under non-radial perturbations. AMS subject classification: 35R35, 35B35, 76D27.