Linear and nonlinear exponential stability of traveling waves for hyperbolic systems with relaxation

Abstract

This paper is concerned with the linear and nonlinear exponential stability of trav- eling wave solutions for a system of quasi-linear hyperbolic equations with relaxation. By applying C0-semigroup theory and detailed spectral analysis, we prove the linear exponential stability of the traveling waves for the quasilinear systems and nonlinear exponential stability of the waves for semi- linear systems, i.e., the Jin-Xin relaxation models, in some exponentially weighted spaces without assuming that the wave strengths are small.