A Parabolic–Hyperbolic Quasilinear System

Abstract

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