Lie Group Action and Stability Analysis of Stationary Solutions for a Free Boundary Problem Modelling Tumor Growth

TL;DR

This paper analyzes the stability of stationary solutions in a tumor growth model using Lie group actions, establishing conditions for stability and instability based on surface tension and diffusion parameters.

Contribution

It introduces a general framework for stability analysis of free boundary problems using Lie group actions without restrictive space assumptions.

Findings

Stable stationary solution when surface tension exceeds threshold and diffusion ratio is small.

Unstable stationary solution when surface tension is below threshold.

Provides a spectral analysis approach for tumor growth models.

Abstract

In this paper we study asymptotic behavior of solutions for a multidimensional free boundary problem modelling the growth of nonnecrotic tumors. We first establish a general result for differential equations in Banach spaces possessing a local Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either $D_A(\theta)$ or $D_A(\theta,\infty)$ type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient $\gamma$ is larger than a threshold value $\gamma^\ast$ then the unique stationary solution is asymptotically stable modulo translations, provided the constant $c$ representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small, whereas if $\gamma< \gamma^\ast$ then this stationary solution is unstable.