Linear stability of the first bifurcation in a tumor growth free boundary problem via local bifurcation structure

TL;DR

This paper analyzes the linear stability of bifurcating solutions in a 3D tumor growth model, revealing instability of non-radial solutions near bifurcation points using bifurcation structure analysis.

Contribution

It introduces a novel approach to assess linear stability of bifurcating solutions in a tumor growth free boundary problem without explicit solution expressions.

Findings

Bifurcation curve exhibits a transcritical bifurcation with negative slope.

Stationary bifurcation solutions are linearly unstable under non-radial perturbations.

Standard stability analysis methods are inadequate due to a complex eigenstructure.

Abstract

In this paper, we consider a 3-dimensional free boundary problem modeling tumor growth with the Robin boundary condition. The system involves a positive parameter $\mu$ which reflects the intensity of tumor aggressiveness. Huang, Zhang and Hu [Nonlinear Anal. Real World Appl. 2017(35), 483-502] have shown that for each $\mu_n$ ($n$ even) in a strictly increasing sequence $\{ \mu_n \}(n\geq 2)$, there exists a stationary bifurcation solution $(\sigma_n(\varepsilon),p_n(\varepsilon),r_n(\varepsilon))$ with $\mu = \mu_n(\varepsilon)$ bifurcating from $\mu_n$. We first derive that the bifurcation curve $(r_2(\varepsilon),\mu_2(\varepsilon))$ exhibits a transcritical bifurcation with $\mu_2'(0)<0$. Moreover, we show that the stationary bifurcation solution $(\sigma_2(\varepsilon),p_2(\varepsilon),r_2(\varepsilon))$ is linearly unstable for small $|\varepsilon|$ under non-radially symmetric perturbations. In contrast to the linear stability of the radially symmetric stationary solution, the lack of explicit expressions for bifurcation solutions adds great difficulty in analyzing their linear stability. The novelty of this paper lies in the use of the bifurcation curve's structure to overcome the above difficulties. Moreover, this linear stability result is not established using the standard method, due to an eight-dimensional generalized kernel at eigenvalue 0 for the linearized operator.