Asymptotic Stability of Solutions to a Hyperbolic-Elliptic Coupled System of Radiating Gas on the Half Line

Abstract

This paper is concerned with the asymptotic stability of the solution to an initial-boundary value problem on the half line for a hyperbolic-elliptic coupled system of radiating gas, where the data on the boundary and at the far field state are defined as $u_-$ and $u_+$ satisfying $u_-<u_+$. For the scalar viscous conservation law case, it is known by the work of Liu, Matsumura, and Nishihara [SIAM J. Math. Anal., 29 (1998), pp. 293--308] that the solution tends toward a rarefaction wave or stationary solution or superposition of these two kinds of waves depending on the distribution of $u_\pm$. Motivated by their work, we prove the stability of the above three types of wave patterns for the hyperbolic-elliptic coupled system of radiating gas with small perturbation. A singular phase plane analysis method is introduced to show the existence and the precise asymptotic behavior of the stationary solution, especially for the degenerate case: $u_-<u_+=0$ such that the system has inevitable singularities. The stability of the rarefaction wave, the stationary solution, and their superposition is proved by applying the standard $L^2$-energy method.