This paper is concerned with the linear and nonlinear exponential stability of traveling 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.
"Linear and nonlinear exponential stability of traveling waves for hyperbolic systems with relaxation." Commun. Math. Sci. 7 (3) 571 - 593, September 2009.