Scaling variables and stability of hyperbolic fronts

We consider the damped hyperbolic equation (1) epsilon u(tt) + u(t) = u(xx) + F(u), x is an element of R, t greater than or equal to 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, ( 1) has a family of monotone traveling wave solutions ( or propagating fronts) connecting the equilibria u = 0 and u = 1. This family is indexed by a parameter c greater than or equal to c(*) related to the speed of the front. In the critical case c = c(*), we prove that the traveling 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 --> +infinity and approach a universal self-similar profile, which is independent of epsilon, F, and the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting epsilon = 0 in ( 1). The proof of our results relies on various energy estimates for (1) rewritten in self-similar variables x/root t, log t.