# A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth

### Avner Friedman

Ohio State University, Columbus, USA

## Abstract

We consider a tumor model with three populations of cells: proliferating, quiescent, and necrotic. Cells may change from one type to another at a rate which depends on the nutrient concentration. We assume that the tumor tissue is a fluid subject to the Stokes equation with sources determined by the proliferation rate of the proliferating cells. The boundary of the tumor is a free boundary held together by cell-to-cell adhesiveness of intensity $γ$. Thus, on the free boundary the stress tensor $T$ and the mean curvature $κ$ are related by $Tn=−γκn$ where $n$ is the outward normal. We prove that the coupled system of PDEs for the densities of the three types of cells, the nutrient concentration, and the fluid velocity and pressure have a unique smooth solution, with a smooth free boundary, for a small time interval.

## Cite this article

Avner Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth. Interfaces Free Bound. 8 (2006), no. 2, pp. 247–261

DOI 10.4171/IFB/142