Decay Rate for Travelling Waves of a Relaxation Model

A relaxation model was proposed in [Shi Jin and Zhouping Xin,Comm. Pure Appl. Math.48(1995), 555–563] to approximate the hyperbolic systems numerically under the subcharacteristic condition introduced in [T. P. Liu,Comm. Math. Phys.108(1987), 153–175]. The stability of travelling waves with strong shock profile and integral zero was proved in [H. L. Liu, J. H. Wang, and T. Yang, Stability in a relaxation model with nonconvex flux, preprint, 1996; H. L. Liu and J. Wang, Asymptotic stability of travelling wave solutions of a hyperbolic system with relaxation terms, preprint, 1995] when the original system is scalar. In this paper, we study the rate of the asymptotic convergence speed of thse travelling wave solutions. The analysis applies to the case of a nonconvex flux and when the shock speed coincides with characteristic speed of the state at infinity. The decay rate is obtained by applying the energy method and is shown to be the same as the one for the viscous conservation law [A. Matsumura and K. Nishihara,Comm. Math. Phys.165(1994), 83–96].