 98756 Th. Gallay and G. Raugel (Paris XI)
 Scaling Variables and Stability of Hyperbolic Fronts
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Dec 18, 98

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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 selfsimilar 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 selfsimilar
variables x/\sqrt{t}, \log t.
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