Stability of traveling wave solutions with time-dependent shifts to a semilinear hyperbolic system with relaxation

In this paper, we consider the stability of traveling wave solutions for a semilinear hyperbolic system with relaxation. We aim to study a new approach to the stability of viscous shock waves instead of the anti-derivative method. We show that the traveling wave solutions with time-dependent shifts for the Jin–Xin model are asymptotically stable under the smallness conditions on the initial perturbations and wave strength. Our approach is based on the results obtained by Kang, Vasseur and Wang.