The derivation from spectral stability of the asymptotic stability (in the sense of Lyapunov, i.e. in large time) of traveling waves of hyperbolic systems is an important question, that is still open to a large extent. Among difficulties to overcome stand three facts
* the systems under consideration are in general quasi-linear whereas the dynamics does not exhibit strong regularization effects;
* wave profiles contain in general characteristic points where, even in dimension 1, the underlying operators lose ellipticity;
* wave profiles may be discontinuous so that the perturbed evolution problem becomes of mixed initial/boundary value type with free surfaces of discontinuity.
In the present talk, based on recent contributions with Vincent Duchêne (CNRS, Rennes), we shall see how those difficulties may be bypassed relatively easily for waves of scalar balance laws.