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

TL;DR

This paper proves the local asymptotic stability of the unique stationary solution for a hyperbolic free boundary tumor growth model involving proliferating and quiescent cells, using advanced functional analysis techniques.

Contribution

It establishes the local asymptotic stability of the stationary solution for a specific tumor growth model through a novel application of semigroup theory and linear estimates.

Findings

Proves local asymptotic stability of the stationary solution.

Utilizes a functional approach and $C_0$ semigroup theory.

Improves linear estimates from previous work.

Abstract

In this paper we study asymptotic behavior of solutions for a free boundary problem modeling the growth of tumors containing two species of cells: proliferating cells and quiescent cells. This tumor model was proposed by Pettet et al in {\em Bull. Math. Biol.} (2001). By using a functional approach and the $C_0$ semigroup theory, we prove that the unique stationary solution of this model ensured by the work of Cui and Friedman ({\em Trans. Amer. Math. Soc.}, 2003) is locally asymptotically stable in certain function spaces. Key techniques used in the proof include an improvement of the linear estimate obtained by the work of Chen et al ({\em Trans. Amer. Math. Soc.}, 2005), and a similarity transformation.