Nonlinear stability of viscous shock profiles for a hyperbolic system with Cattaneo’s law and non-convex flux

Abstract

This paper is concerned with the large time behavior of solutions to the scalar conservation law with an artificial heat flux term. The heat flux is governed by Cattaneo’s law, which leads to a 2 × 2 system of hyperbolic equations. The existence and nonlinear stability of rarefaction waves and viscous shock waves have been derived under the assumption that flux function is strictly convex. In the current paper, we focus on the one-dimensional Cauchy problem for the system which allows for non-convex flux. Under Oleinik entropy condition, we obtain the existence and asymptotic stability of shifted viscous shock waves with sufficiently small wave strength. The proof is based on the standard energy method and shift theory.