Analysis of a mathematical model for the growth of cancer cells

In this paper, a two-dimensional model for the growth of multi-layer tumors is presented. The model consists of a free boundary problem for the tumor cell membrane and the tumor is supposed to grow or shrink due to cell proliferation or cell dead. The growth process is caused by a diffusing nutrient concentration $\sigma$ and is controlled by an internal cell pressure $p$. We assume that the tumor occupies a strip-like domain with a fixed boundary at $y=0$ and a free boundary $y=\rho(x)$, where $\rho$ is a $2\pi$-periodic function. First, we prove the existence of solutions $(\sigma,p,\rho)$ and that the model allows for peculiar stationary solutions. As a main result we establish that these equilibrium points are locally asymptotically stable under small perturbations.