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 s is the pressure. The pressure is a result of the transport of cells which proliferate or die. The motion of the free boundary partial derivativeOmega(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 less than or equal to t less than or equal to T.