Asymptotic stability for a free boundary tumor model with angiogenesis

TL;DR

This paper investigates the nonlinear stability of a free boundary tumor growth model with angiogenesis, proving that the radially symmetric stationary solution remains stable under certain conditions, despite mathematical challenges.

Contribution

It introduces a new fixed point theorem to analyze nonlinear stability, addressing the difficulty of unknown sphere centers due to mode 1 perturbations.

Findings

Radially symmetric stationary solution is nonlinearly stable for 0<μ<μ*

Develops a new fixed point theorem for stability analysis

Addresses mathematical challenges from mode 1 perturbations

Abstract

In this paper, we study a free boundary problem modeling solid tumor growth with vasculature which supplies nutrients to the tumor; this is characterized in the Robin boundary condition. It was recently established [Discrete Cont. Dyn. Syst. 39 (2019) 2473-2510] that for this model, there exists a threshold value $\mu^\ast$ such that the unique radially symmetric stationary solution is linearly stable under non-radial perturbations for $0<\mu<\mu^\ast$ and linearly unstable for $\mu>\mu^\ast$. In this paper we further study the nonlinear stability of the radially symmetric stationary solution, which introduces a significant mathematical difficulty: the center of the limiting sphere is not known in advance owing to the perturbation of mode 1 terms. We prove a new fixed point theorem to solve this problem, and finally obtain that the radially symmetric stationary solution is nonlinearly stable for $0<\mu<\mu^\ast$ when neglecting translations.