Nonlocalised damping estimates for hyperbolic relaxation systems in one space dimensions

In this paper, we present a new approach to obtain so-called damping estimates for self-similar solutions to general hyperbolic relaxation systems applying the method of characteristics. Such damping estimates are an important part of the stability theory of shock profiles, where they enable the closure of nonlinear stability arguments. We extend the damping estimates obtained in Mascia and Zumbrun (2005) from the $L^2$-case to the $L^{\mathrm{\infty }}$-case and, at the same time, generalize the $L^2$-estimates to the non-symmetric setting. Our estimates open the door to a general stability theory of shock profiles of hyperbolic relaxation systems under nonlocalised perturbations in one space dimension.