Nonlinear stability of viscous shock waves and rarefaction waves for a hyperbolic system with Cattaneo’s law under periodic perturbation

Abstract This paper is concerned with the time-asymptotically nonlinear stability of viscous shock waves and rarefaction waves to the Cauchy problem for a scalar conservation law with an artificial heat flux satisfying Cattaneo’s law under space-periodic perturbations. Since the heat flux equation is not of conservation type, and the average of periodic solutions to which decays in time, we need to modify the ansatz when dealing with the heat flux. Based on the energy method, it is proved that: if the initial perturbation and the strength of shock wave or rarefaction wave are small enough, then the solution time-asymptotically stable to a viscous shock wave with a constant shift or a rarefaction wave respectively. And the constant shift in the viscous shock wave depends on both the periodic perturbation and the mass of the localized perturbation.