  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 the 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) 293-308) that the
solution tends toward rarefaction wave or stationary solution or superposition
of these two kind 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 the 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 rarefaction wave, stationary
solution, and their superposition, is proved by applying the standard
$L^2$-energy method.
