Existence and asymptotic stability of stationary waves for symmetric hyperbolic-parabolic systems in half-line.

In this paper, we consider a one-dimensional half-space problem for a system of viscous conservation laws which is deduced to a symmetric hyperbolic-parabolic system under assuming that the system has a strictly convex entropy function. We firstly prove existence of a stationary solution by assuming that a boundary strength is sufficiently small. The existence of the stationary solution is characterized by the number of negative characteristics. In the case where one characteristic speed is zero at spatial asymptotic state , we assume that the characteristic field corresponding to the characteristic speed 0 is genuinely nonlinear in order to show existence of a degenerate stationary solution with the aid of a center manifold theory. We next prove that the stationary solution is time asymptotically stable under a smallness assumption on an initial perturbation in the Sobolev space. The key to proof is to derive the uniform a priori estimates by using the energy method, where the stability condition of Shizuta-Kawashima type plays an essential role. [ABSTRACT FROM AUTHOR]