Symmetry-breaking bifurcation for necrotic tumor model with two free boundaries

Abstract

In this paper, we study a 2-dimensional free boundary problem modeling the tumor growth with a necrotic core. This model has three parameters: ${\sigma }_D$ is a threshold value of nutrient concentration for distinguishing whether tumor cells are alive or not, $\tilde{\sigma }$ is the death rate of proliferating cells and $\nu$ is the removal rate of necrotic cells. With the assumption of $\tilde{\sigma }-{\sigma }_D-\nu \le 0$, we first give a complete classification of ${\sigma }_D$ and $\tilde{\sigma }$ under which the necrotic problem either has the unique radially symmetric stationary solution $\left({\sigma }_s,p_s,{\rho }_s,R_s\right)$ or no solutions. Furthermore, we derive the existence of symmetry-breaking solutions bifurcating from the radially symmetric solution $\left({\sigma }_s,p_s,{\rho }_s,R_s\right)$ for every ${\gamma }_l$ $(l=2,3,\dots )$.