American Institute of Mathematical Sciences

May  2019, 39(5): 2473-2510. doi: 10.3934/dcds.2019105

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

 1 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China 2 Department of Applied Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

* Corresponding author: Zhengce Zhang

Received  January 2018 Revised  October 2018 Published  January 2019

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.

Citation: Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105
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