A Free Boundary Problem for an Elliptic-Hyperbolic System: An Application to Tumor Growth

We consider a system of two hyperbolic equations for p,q and two elliptic equations for $c,\sigma$, where p,q are the densities of cells within the tumor $\Omega_t$ in proliferating and quiescent states, respectively, c is the concentration of nutrients, and $\sigma$ is the pressure. The pressure is a result of the transport of cells which proliferate or die. The motion of the free boundary $\partial \Omega_t$ is given by the continuity condition, and $\sigma$ at the free boundary is proportional to the surface tension. We prove the existence, uniqueness, and regularity of the solution for a small time interval $0\leq t\leq T$.

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