Convergence rates toward the planar stationary solution for a hyperbolic-elliptic coupled system with boundary effect corresponding to shock wave

In this paper, we study the asymptotic behavior of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupled system of the radiating gas on half space with the conditions $u(0,y,t)=u_{-}$ and $u(\infty ,y,t)=u_{+}<u_{-}$, where the corresponding Cauchy problem admits the shock wave as an asymptotic profile. In the case of $u_{+}<u_{-}\le 0$, we prove that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity by assuming that the initial perturbation is small. Furthermore, we obtain the convergence rate by applying the time and space weighted energy method. The results include one-dimensional and two-dimensional cases.