Asymptotic Behavior of Solutions of a Free Boundary Problem Modelling the Growth of Tumors with Stokes Equations

TL;DR

This paper analyzes a free boundary problem modeling tumor growth with fluid dynamics governed by Stokes equations, establishing conditions for stability of radially symmetric solutions based on surface tension effects.

Contribution

It introduces a novel stability analysis of tumor growth models with fluid-like tissues using a functional approach and identifies a critical surface tension threshold for stability.

Findings

Existence of a unique radially symmetric stationary solution.

Stability of the solution depends on the surface tension coefficient.

Identification of a critical threshold for stability based on surface tension.

Abstract

We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value $\gamma_*>0$ for the surface tension coefficient $\gamma$, such that in the case $\gamma>\gamma_*$ this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable.