Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Thierry Gallay; Romain Joly

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 1, page 103-140
  • ISSN: 0012-9593

Abstract

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We consider the damped wave equation α u t t + u t = u x x - V ' ( u ) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u ( x , t ) = h ( x - s t ) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V . We show that, if the initial data are sufficiently close to the profile of a front for large | x | , the solution of the damped wave equation converges uniformly on ℝ to a travelling front as t → + ∞ . The proof of this global stability result is inspired by a recent work of E. Risler [38] and relies on the fact that our system has a Lyapunov function in any Galilean frame.

How to cite

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Gallay, Thierry, and Joly, Romain. "Global stability of travelling fronts for a damped wave equation with bistable nonlinearity." Annales scientifiques de l'École Normale Supérieure 42.1 (2009): 103-140. <http://eudml.org/doc/272207>.

@article{Gallay2009,
abstract = {We consider the damped wave equation $\alpha u_\{tt\}+u_t=u_\{xx\}-V^\{\prime \}(u)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u(x,t)=h(x-st)$ which describe a moving interface between two different steady states of the system, one of which being the global minimum of $V$. We show that, if the initial data are sufficiently close to the profile of a front for large $|x|$, the solution of the damped wave equation converges uniformly on $\mathbb \{R\}$ to a travelling front as $t \rightarrow +\infty $. The proof of this global stability result is inspired by a recent work of E. Risler [38] and relies on the fact that our system has a Lyapunov function in any Galilean frame.},
author = {Gallay, Thierry, Joly, Romain},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {travelling front; global stability; damped wave equation; Lyapunov function},
language = {eng},
number = {1},
pages = {103-140},
publisher = {Société mathématique de France},
title = {Global stability of travelling fronts for a damped wave equation with bistable nonlinearity},
url = {http://eudml.org/doc/272207},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Gallay, Thierry
AU - Joly, Romain
TI - Global stability of travelling fronts for a damped wave equation with bistable nonlinearity
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 1
SP - 103
EP - 140
AB - We consider the damped wave equation $\alpha u_{tt}+u_t=u_{xx}-V^{\prime }(u)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u(x,t)=h(x-st)$ which describe a moving interface between two different steady states of the system, one of which being the global minimum of $V$. We show that, if the initial data are sufficiently close to the profile of a front for large $|x|$, the solution of the damped wave equation converges uniformly on $\mathbb {R}$ to a travelling front as $t \rightarrow +\infty $. The proof of this global stability result is inspired by a recent work of E. Risler [38] and relies on the fact that our system has a Lyapunov function in any Galilean frame.
LA - eng
KW - travelling front; global stability; damped wave equation; Lyapunov function
UR - http://eudml.org/doc/272207
ER -

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