Asymptotic Stability of the Stationary Solution for a Parabolic-Hyperbolic Free Boundary Problem Modeling Tumor Growth

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 quiescent cells. In previous literature 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 $0<\varepsilon\ll 1$. 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 $0<\varepsilon\ll 1$.