Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth

TL;DR

This paper proves the asymptotic stability of a stationary tumor growth model with two cell types for small positive parameters, extending previous results from the limit case to more realistic scenarios.

Contribution

It demonstrates the stability of the stationary solution in a coupled parabolic-hyperbolic system for small positive epsilon, using variable and similarity transforms.

Findings

Stationary solution remains asymptotically stable for small epsilon

Extended stability results from limit case to realistic model

Applied transform techniques to analyze complex coupled system

Abstract

This paper studies asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with two species of cells: proliferating cells and quiecent cells. In previous literatures it has been proved that this problem has a unique stationary solution which is asymptotically stable in the limit case $\varepsilon=0$. In this paper we consider the more realistic case $0<\varepsilon<<1$. In this case, after suitable reduction the model takes the form of a coupled system of a parabolic equation and a hyperbolic system, so that it is more difficult than the limit case $\varepsilon=0$. By using some unknown variable transform as well as the similarity transform technique developed in our previous work, we prove that the stationary solution is also asymptotically stable in the case $0<\varepsilon<<1$.