Scaling Variables and Stability of Hyperbolic Fronts

• Published in: SIAM journal on mathematical analysis 2000, Vol.32 (1), p.1-29
• Main Authors: Gallay, Thierry, Raugel, Geneviève
• Format: Article
• Language: English
• Subjects: Variables
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/S0036141099351334

We consider the damped hyperbolic equation $$ \epsilon u_{tt} + u_t \,=\, u_{xx} + \FF(u) , \quad x \in \mathbf{R} , \quad t \ge 0 , \leqno(1) $$ where $\epsilon$ is a positive, not necessarily small parameter. We assume that $\FF(0) = \FF(1) = 0$ and that $\FF$ 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 $c \ge c_*$ 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 $t \to +\infty$ 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 $x/\sqrt{t}$, $\log t$.