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

TL;DR

This work analyzes a 3D tumor-growth free boundary problem with Robin boundary conditions and a controlling parameter $\mu$ that measures tumor aggressiveness. The authors establish that the first nonradial bifurcation branch from the radially symmetric state is transcritical with $\mu_2'(0)<0$, and they prove the corresponding axial-symmetric stationary solution is linearly unstable to nonradial perturbations for small $|\varepsilon|$. A key novelty is exploiting the local bifurcation structure to deduce stability properties without explicit bifurcating profiles, overcoming the challenge of an $8$-dimensional $0$-group kernel in the linearized operator. The analysis combines a precise expansion of the boundary geometry, spectral decomposition via spherical harmonics, and Kato-type perturbation theory to track the $0$-group eigenvalues along the bifurcation curve. The results extend previous radial-stability analyses by addressing the stability of stationary bifurcation solutions in the nonradial setting and highlight the critical role of the bifurcation-curve structure in understanding stability.

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 $μ$ 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 $μ_n$ ($n$ even) in a strictly increasing sequence $\{ μ_n \}(n\geq 2)$, there exists a stationary bifurcation solution $(σ_n(\varepsilon),p_n(\varepsilon),r_n(\varepsilon))$ with $μ= μ_n(\varepsilon)$ bifurcating from $μ_n$. We first derive that the bifurcation curve $(r_2(\varepsilon),μ_2(\varepsilon))$ exhibits a transcritical bifurcation with $μ_2'(0)<0$. Moreover, we show that the stationary bifurcation solution $(σ_2(\varepsilon),p_2(\varepsilon),r_2(\varepsilon))$ 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.

Figures (1)

• Figure 1: Linear Stability

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1
• proof
• Proposition 4.2
• Remark 4.3