Th. Gallay and G. Raugel (Paris XI)
Scaling Variables and Stability of Hyperbolic Fronts
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ABSTRACT.  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.