Scaling Variables and Stability of Hyperbolic Fronts

We consider the damped hyperbolic equation (1) \epsilon u_{tt} + u_t = u_{xx} + F(u), x \in R, t \ge 0, where \epsilon is a positive, not necessarily small parameter. We assume that F(0) = F(1) = 0 and that F is concave on the interval [0,1]. Under these hypotheses, Eq.(1) has a family of monotone travelling wave solutions (or propagating fronts) connecting the equilibria u=0 and u=1. This family is indexed by a parameter c \ge c_* related to the speed of the front. In the critical case c=c_*, we prove that the travelling wave is asymptotically stable with respect to perturbations in a weighted Sobolev space. In addition, we show that the perturbations decay to zero like t^{-3/2} as t \to +\infty and approach a universal self-similar profile, which is independent of \epsilon, F and of the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting \epsilon = 0 in Eq.(1). The proof of our results relies on careful energy estimates for the equation (1) rewritten in self-similar variables x/\sqrt{t}, \log t.