Stability of Propagating Fronts in Damped Hyperbolic Equations
| dc.creator | Gallay, Th. | |
| dc.creator | Raugel, G. | |
| dc.date | 1998-09-18 | |
| dc.date.accessioned | 2026-07-07T05:43:26Z | |
| dc.date.available | 2026-07-07T05:43:26Z | |
| dc.description | We consider the damped hyperbolic equation in one space dimension $εu_{tt} + u_t = u_{xx} + F(u)$, where $ε$ 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 $ε= 0$, we recover several stability results for the travelling waves of the corresponding parabolic equation. | |
| dc.description | 20 pages, plain TeX | |
| dc.identifier | https://arxiv.org/abs/patt-sol/9809007 | |
| dc.identifier | http://arxiv.org/abs/patt-sol/9809007 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/83308 | |
| dc.subject | Pattern Formation and Solitons | |
| dc.title | Stability of Propagating Fronts in Damped Hyperbolic Equations | |
| dc.type | text |