Stability of viscous shock waves associated with non-convex modes

In this paper we consider small-amplitude viscous shock waves associated wi th a possibly non-convex mode (i.e., a not necessarily "genuinely nonlinear field") and prove their time-asymptotic stability towards sufficiently sma ll perturbations of not necessarily vanishing mass. We thus extend the resu lt of [SX] to the case of non-convex modes and the result of [F1] to the ca se of non-zero mass perturbations.