Article Contents
Article Contents

# Bifurcation from stability to instability for a free boundary tumor model with angiogenesis

• * Corresponding author: Zhengce Zhang
• In this paper we consider a free boundary tumor model with angiogenesis. The model consists of a reaction-diffusion equation describing the concentration of nutrients $\sigma$ and an elliptic equation describing the distribution of the internal pressure $p$. The vasculature supplies nutrients to the tumor, so that $\frac{\partial\sigma}{\partial \mathbf{n}}+\beta(\sigma-\bar{\sigma}) = 0$ holds on the boundary, where a positive constant $\beta$ is the rate of nutrient supply to the tumor and $\bar{\sigma}$ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate $\mu$. If $0<\widetilde{\sigma}<\overline{\sigma}$, where $\widetilde{\sigma}$ is a threshold concentration for proliferating, then there exists a unique radially symmetric stationary solution $(\sigma_S(r), p_S(r), R_S)$. In this paper, we found a function $\mu^\ast = \mu^\ast(R_S)$ such that if $\mu<\mu^\ast$ then the radially symmetric stationary solution is linearly stable with respect to non-radially symmetric perturbations, whereas if $\mu>\mu^\ast$ then the radially symmetric stationary solution is linearly unstable.

Mathematics Subject Classification: Primary: 35B35, 35R35, 35B40, 35C10, 92C37.

 Citation:

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