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

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 sigma and inner pressure p within the tumor tissue. Angiogenesis results in a new boundary condition partial derivative(n sigma) + beta (sigma - (sigma) over bar) = 0 instead of the widely studied condition sigma = (sigma) over bar over the moving boundary, where beta 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 gamma(*) is obtained such that the radial stationary solution is asymptotically stable for gamma > gamma(*) and unstable for 0 < gamma < gamma(*). (C) 2018 Elsevier Ltd. All rights reserved.