# American Institute of Mathematical Sciences

## Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core

 1 College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China 2 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

Received  September 2019 Revised  November 2019 Published  January 2020

In this paper, we present a rigorous mathematical analysis of a free boundary problem modeling the growth of a vascular solid tumor with a necrotic core. If the vascular system supplies the nutrient concentration $\sigma$ to the tumor at a rate $\beta$, then $\frac{\partial\sigma}{\partial\bf n}+\beta(\sigma-\bar\sigma) = 0$ holds on the tumor boundary, where $\bf n$ is the unit outward normal to the boundary and $\bar\sigma$ is the nutrient concentration outside the tumor. The living cells in the nonnecrotic region proliferate at a rate $\mu$. We show that for any given $\rho>0$, there exists a unique $R\in(\rho, \infty)$ such that the corresponding radially symmetric solution solves the steady-state necrotic tumor system with necrotic core boundary $r = \rho$ and outer boundary $r = R$; moreover, there exist a positive integer $n^{**}$ and a sequence of $\mu_n$, symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solution for each $\mu_n$ (even $n\ge n^{**})$.

Citation: Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020084
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