# Interfaces and Free Boundaries

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**Volume 8, Issue 2, 2006, pp. 247–261**

**DOI: 10.4171/IFB/142**

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

Avner Friedman^{[1]}(1) Mathematical Biosciences Institute, Ohio State University, 231 18th Avenue West, OH 43210, COLUMBUS, UNITED STATES

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 $\gamma$. Thus, on the free boundary the stress tensor $T$ and the mean curvature $\kappa$ are related by $T\vec n=-\gamma\kappa\vec n$ where $\vec n$ 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.

*Keywords: *Tumor growth, free boundary problems, hyperbolic equations, Stokes equation

Friedman Avner: A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth. *Interfaces Free Bound.* 8 (2006), 247-261. doi: 10.4171/IFB/142