Asymptotic behavior of solutions of a free-boundary tumor model with angiogenesis

Abstract

This paper is concerned with a free boundary problem modeling the growth of solid tumor spheroid with angiogenesis. The model comprises a coupled system of two elliptic equations describing the distribution of nutrient concentration σ and inner pressure p within the tumor tissue. Angiogenesis results in a new boundary condition ∂nσ+βσ−σ¯=0 instead of the widely studied condition σ=σ¯ over the moving boundary, where β is a positive constant. We first prove that this problem admits a unique radial stationary solution, and this solution is globally asymptotically stable under radial perturbations. Then we establish local well-posedness of the problem and study asymptotic stability of the radial stationary solution under non-radial perturbations. A positive threshold value γ∗ is obtained such that the radial stationary solution is asymptotically stable for γ>γ∗ and unstable for 0<γ<γ∗.