Shock waves for a model system of the radiating gas

This paper is concerned with the existence and the asymptotic stability of traveling waves for a model system derived from approximating the one-dimensional system of the radiating gas. We show the existence of smooth or discontinuous traveling waves and also prove the uniqueness of these traveling waves under the entropy condition, in the class of piecewise smooth functions with the first kind discontinuities. Furthermore, we show that the C³-smooth traveling waves are asymptotically stable and that the rate of convergence toward these waves is t^-1/4, which looks optimal. The proof of stability is given by applying the standard energy method to the integrated equation of the original one.