Scaling Variables and Stability of Hyperbolic Fronts
| dc.creator | Gallay, Th. | |
| dc.creator | Raugel, G. | |
| dc.date | 1998-12-18 | |
| dc.date.accessioned | 2026-07-07T05:43:28Z | |
| dc.date.available | 2026-07-07T05:43:28Z | |
| dc.description | We consider the damped hyperbolic equation (1) εu_{tt} + u_t = u_{xx} + F(u), x \in R, t \ge 0, where εis 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 ε, F and of the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting ε= 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. | |
| dc.description | 38 pages, plain TeX | |
| dc.identifier | https://arxiv.org/abs/patt-sol/9812007 | |
| dc.identifier | http://arxiv.org/abs/patt-sol/9812007 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/83316 | |
| dc.subject | Pattern Formation and Solitons | |
| dc.title | Scaling Variables and Stability of Hyperbolic Fronts | |
| dc.type | text |