Asymptotic stability for a free boundary tumor model with a periodic supply of external nutrients

TL;DR

This paper proves the asymptotic stability of a spherical tumor model with periodic nutrient supply, extending previous linear stability results to the fully nonlinear case, and highlighting the role of mitosis rate in tumor morphology.

Contribution

It establishes the asymptotic stability of the spherical tumor solution for the nonlinear model when the mitosis parameter is below a critical threshold, building on prior linear stability findings.

Findings

Spherical tumor solution is asymptotically stable for certain parameters.

The model incorporates periodic external nutrient supply.

Stability depends on the mitosis parameter .

Abstract

For tumor growth, the morphological instability provides a mechanism for invasion via tumor fingering and fragmentation. This work considers the asymptotic stability of a free boundary tumor model with a periodic supply of external nutrients. The model consists of two elliptic equations describing the concentration of nutrients and the distribution of the internal pressure in the tumor tissue, respectively. The effect of the parameter $\mu$ representing a measure of mitosis on the morphological stability are taken into account. It was recently established in [25] that there exists a critical value $\mu_\ast$ such that the unique spherical periodic positive solution is linearly stable for $0<\mu<\mu_\ast$ and linearly unstable for $\mu>\mu_\ast$. In this paper, we further prove that the spherical periodic positive solution is asymptotically stable for $0<\mu<\mu_\ast$ for the fully nonlinear problem.