dorsal/arxiv
View SchemaScaling Variables and Stability of Hyperbolic Fronts
| Authors | Th. Gallay, G. Raugel |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9812007 |
| URL | https://arxiv.org/abs/patt-sol/9812007 |
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.
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"abstract": "We consider the damped hyperbolic equation (1) \\epsilon u_{tt} + u_t = u_{xx}\n+ F(u), x \\in R, t \\ge 0, where \\epsilon is a positive, not necessarily small\nparameter. We assume that F(0) = F(1) = 0 and that F is concave on the interval\n[0,1]. Under these hypotheses, Eq.(1) has a family of monotone travelling wave\nsolutions (or propagating fronts) connecting the equilibria u=0 and u=1. This\nfamily is indexed by a parameter c \\ge c_* related to the speed of the front.\nIn the critical case c=c_*, we prove that the travelling wave is asymptotically\nstable with respect to perturbations in a weighted Sobolev space. In addition,\nwe show that the perturbations decay to zero like t^{-3/2} as t \\to +\\infty and\napproach a universal self-similar profile, which is independent of \\epsilon, F\nand of the initial data. In particular, our solutions behave for large times\nlike those of the parabolic equation obtained by setting \\epsilon = 0 in\nEq.(1). The proof of our results relies on careful energy estimates for the\nequation (1) rewritten in self-similar variables x/\\sqrt{t}, \\log t.",
"arxiv_id": "patt-sol/9812007",
"authors": [
"Th. Gallay",
"G. Raugel"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"title": "Scaling Variables and Stability of Hyperbolic Fronts",
"url": "https://arxiv.org/abs/patt-sol/9812007"
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