Stability analysis of a multi-layer tumor model with free boundary

In this paper we study a multi-layer tumor model which is expressed as a free boundary problem of a system of partial differential equations. The problem consists of two elliptic equations describing the distribution of nutrient concentration and the pressure between tumor cells, respectively, in an unbounded strip-like region in which the tumor occupies. This region has two disjoint boundaries: While the lower part is fixed, the upper part, which stands for the tumor surface, can move as the tumor grows. Under certain conditions, the problem can be proved to admit a unique equilibrium which corresponds to the flat upper boundary. We first convert the model into a parabolic differential equation in certain function space. Next we compute the spectrum of the linearized problem at the equilibrium. By applying the geometric theory of parabolic differential equations in Banach spaces, we prove that if the cell-to-cell adhesiveness coefficient $\gamma$ is larger than a threshold value ${\gamma }^1$, then the unique flat equilibrium is asymptotically stable, whereas in the case $0<\gamma <{\gamma }^1$ the flat equilibrium is unstable.