A parabolic-hyperbolic quasilinear system

We prove the existence and uniqueness of solutions of the parabolic-hyperbolic system u(t+)div(u del phi) = G(u,phi), phi(t) -Delta phi = F(u, phi) in Omega x (0, infinity) for unknown (u, phi) = (u(x, t), phi(x, t)), where Omega is a bounded domain in R-n with smooth boundary. The system is solved subject to no-flux boundary condition for phi and given initial conditions for u and phi Structural conditions on F and G are assumed to ensure L-infinity a priori estimates on u and phi. The key for global in time existence is the Lipschitz continuity del phi of in the spatial variable x, an intrinsic requirement for the existence of classical solutions to the hyperbolic equation u(t) + del phi.del u = Q. A new method, worked specifically for hyperbolic-parabolic or hyperbolic-elliptic systems, is developed here to establish an L-infinity bound for the Hessian D-x(2)phi. In the final part of the paper we prove the asymptotic stability of stationary solutions.