Asymptotic stability of travelling wave to a hyperbolic-elliptic coupled system of the radiating gas on half line

This paper studies the asymptotic behavior of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupled system of the radiating gas on half line, where the data on the boundary and at the far field are prescribed as u_- and u_+ satisfying u_+0 , we prove that the solution to the problem converges to the properly shifted travelling wave U(x-st+d(t)) as time tends to infinity under small initial perturbation, where d(t) is first given in investigating the scalar viscous conservation laws by Liu and Nishihara (J Differ Equ 133:296–320, 1997). The proof is based on the energy method. The algebraic convergence rate is also given by applying the time weighted energy method.