Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion

This paper is concerned with the stability of traveling front solutions for 2×2 quasi-linear relaxation systems with small diffusion rate. By applying geometric singular perturbation method, special Evans function estimates, detailed spectral analysis and C 0 semigroup theories, we prove that all the non-degenerate waves for semi-linear relaxation systems are locally exponentially stable in some exponentially weighted spaces. We also obtain the linear exponential stability of the non-degenerate waves for quasi-linear relaxation systems, where the wave strengths can be large.