Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities

The present paper is concerned with the asymptotic behavior of a discontinuous solution to a model system of radiating gas. As we assume that an initial data has a discontinuity only at one point, so does the solution. Here the discontinuous solution is supposed to satisfy an entropy condition in the sense of Kruzkov. Previous researches have shown that the solution converges uniformly to a traveling wave if an initial perturbation is integrable and is small in the suitable Sobolev space. If its anti-derivative is also integrable, the convergence rate is known to be (1+t)^-1/4 as time t tends to infnity. In the present paper, we improve the previous result and show that the convergence rate is exactly the same as the spatial decay rate of the initial perturbation.