dorsal/arxiv
View SchemaStability of Propagating Fronts in Damped Hyperbolic Equations
| Authors | Th. Gallay, G. Raugel |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9809007 |
| URL | https://arxiv.org/abs/patt-sol/9809007 |
Abstract
We consider the damped hyperbolic equation in one space dimension $\epsilon u_{tt} + u_t = u_{xx} + F(u)$, 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 assumptions, our equation has a continuous family of monotone propagating fronts (or travelling waves) indexed by the speed parameter $c \ge c_*$. Using energy estimates, we first show that the travelling waves are locally stable with respect to perturbations in a weighted Sobolev space. Then, under additional assumptions on the non-linearity, we obtain global stability results using a suitable version of the hyperbolic Maximum Principle. Finally, in the critical case $c = c_*$, we use self-similar variables to compute the exact asymptotic behavior of the perturbations as $t \to +\infty$. In particular, setting $\epsilon = 0$, we recover several stability results for the travelling waves of the corresponding parabolic equation.
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"abstract": "We consider the damped hyperbolic equation in one space dimension $\\epsilon\nu_{tt} + u_t = u_{xx} + F(u)$, where $\\epsilon$ is a positive, not necessarily\nsmall parameter. We assume that $F(0)=F(1)=0$ and that $F$ is concave on the\ninterval $[0,1]$. Under these assumptions, our equation has a continuous family\nof monotone propagating fronts (or travelling waves) indexed by the speed\nparameter $c \\ge c_*$. Using energy estimates, we first show that the\ntravelling waves are locally stable with respect to perturbations in a weighted\nSobolev space. Then, under additional assumptions on the non-linearity, we\nobtain global stability results using a suitable version of the hyperbolic\nMaximum Principle. Finally, in the critical case $c = c_*$, we use self-similar\nvariables to compute the exact asymptotic behavior of the perturbations as $t\n\\to +\\infty$. In particular, setting $\\epsilon = 0$, we recover several\nstability results for the travelling waves of the corresponding parabolic\nequation.",
"arxiv_id": "patt-sol/9809007",
"authors": [
"Th. Gallay",
"G. Raugel"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"title": "Stability of Propagating Fronts in Damped Hyperbolic Equations",
"url": "https://arxiv.org/abs/patt-sol/9809007"
},
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