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

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.