Convergence to Traveling Waves with Decay Rates for Solutions of the Initial Boundary Problem to a Relaxation Model

In this paper we consider a 2×2 relaxation hyperbolic system of conservation laws with a boundary effect, and we show that the solutions of this initial boundary problem tend to the traveling wave solutions of the corresponding Cauchy problem time-asymptotically. In particular, we give the algebraic and exponential decay rates by using the weighted energy method. The location of a shift for the traveling wave, to overcome the difficulty in the boundary, plays a key role in this paper