Asymptotic stability of planar stationary solution to a 2D hyperbolic-elliptic coupled system of the radiating gas in half space

Abstract: This paper is concerned with the asymptotic stability of planar stationary solution to an initial-boundary value problem for a two-dimensional hyperbolic-elliptic coupled system of the radiating gas in half space. We show that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity under small initial perturbation. These results are proved by the standard $L^2$-energy method. Moreover, we prove that the solution $(u,q)$ converges to the corresponding planar stationary solution at the rate $t^{-\alpha/2-1/4}$ for non-degenerate case, and $t^{-1/4}$ for degenerate case. The proof is based on the time and space weighted energy method.