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

► We study 2×2 quasi-linear relaxation systems with small diffusion rate. ► The main interest is the stability of large-amplitude non-degenerate waves. ► For some quasilinear cases the waves are shown linearly exponentially stable. ► For semilinear cases all the waves are shown nonlinearly exponentially stable. ► The existence and approximation of the wave fronts are also obtained.