Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation

Abstract

In this paper we study a free boundary problem modeling the growth of solid tumor spheroid. It consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumor, respectively. The new feature is that nutrient concentration on the boundary is less than external supply due to a Gibbs-Thomson relation and the problem has two radial stationary solutions, which differs from widely studied tumor spheroid model with surface tension effect. We first establish local well-posedness by using a functional approach based on Fourier multiplier method and analytic semigroup theory. Then we investigate stability of each radial stationary solution. By employing a generalized principle of linearized stability, we prove that the radial stationary solution with a smaller radius is always unstable, and there exists a positive threshold value $\gamma_*$ of cell-to-cell adhesiveness $\gamma$, such that the radial stationary solution with a larger radius is asymptotically stable for $\gamma>\gamma_*$, and unstable for $0<\gamma<\gamma_*$.