Analysis of a tumor-model free boundary problem with a nonlinear boundary condition

Abstract

In this paper we study a free boundary problem modeling the growth of nonnecrotic tumors with angiogenesis. This model differs from the other tumor models studied in existing literatures at the point that it has a nonlinear boundary value condition for the nutrient concentration. We first study spherically symmetric version of this model. We prove that there exists a unique spherically symmetric stationary solution which is asymptotically stable under spherically symmetric perturbation. Next we make rigorous analysis to the spherically asymmetric version of this model. By using some abstract theory of parabolic differential equations in Banach manifold, we prove that this free boundary problem is locally well-posed in little Hölder spaces and the radial stationary solution is asymptotically stable in case the surface tension coefficient γ is larger than a threshold value, whereas unstable in case γ is less than this threshold value.