Convergence rates to the planar stationary solution to a 2D model of the radiating gas on half space

Abstract

This paper is concerned with the asymptotic stability of a planar stationary solution to an initial-boundary value problem for a two-dimensional hyperbolic–elliptic coupled system of the radiating gas on 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. This result is proved by the standard L2-energy method and the div–curl decomposition. Moreover, we prove that the solution (u, q) converges to the corresponding planar stationary solution at the rate t−α/2−1/4 for the non-degenerate case and t−1/4 for the degenerate case. The proof is based on the time and space weighted energy method.