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

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.