Analysis of a Free Boundary Problem Modeling the Growth of Spherically Symmetric Tumors with Angiogenesis

Abstract

This paper is concerned with a free boundary problem modeling the growth of a spherically symmetric tumor with angiogenesis. The unknown nutrient concentration $\sigma =\sigma (r,t)$ occupies the unknown tumor region $r< R(t)$ and satisfies a nonlinear reaction diffusion equation, and the unknown tumor radius $R=R(t)$ satisfies a nonlinear integro-differential equation. Unlike existing literatures on this topic where Dirichlet boundary condition for $\sigma $ is imposed, in this paper the model uses the Robin boundary condition for $\sigma $ . We prove existence and uniqueness of a global in-time classical solution ( $\sigma (r,t),R(t)$ ) for arbitrary $c>0$ and establish asymptotic stability of the unique stationary solution ( $\sigma _{s}(r),R_{s}$ ) for sufficiently small $c$ , where $c$ is a positive constant reflecting the ratio between nutrient diffusion scale and the tumor cell-doubling scale.