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

Gallay, Thierry; Joly, Romain

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity
[Stabilité globale des ondes progressives pour une équation hyperbolique amortie avec non-linéarité bistable]

Gallay, Thierry ; Joly, Romain

Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009), p. 103-140 / Harvested from Numdam

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Abstract

Nous étudions l’équation hyperbolique amortie $α u_{t t} + u_{t} = u_{x x} - V^{'} (u)$ sur la droite réelle, où $V$ est un potentiel bistable. Cette équation possède des ondes progressives de la forme $u (x, t) = h (x - s t)$ qui décrivent le mouvement d’une interface séparant deux états d’équilibre du système, dont l’un est le minimum global de $V$ . Nous montrons que, si les données initiales sont suffisamment proches du profil du front pour $| x |$ grand, alors la solution de l’équation hyperbolique amortie converge uniformément sur $ℝ$ vers une onde progressive lorsque $t \to + \infty$ . La démonstration de ce résultat de stabilité globale s’inspire d’un travail récent de E. Risler [38] et repose sur l’existence pour notre système d’une fonction de Lyapunov dans tout référentiel en translation uniforme.

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 $t \to + \infty$ . 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.

Publié le : 2009-01-01

MR 2518894 | Zbl 1169.35041 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/asens.2091
Classification: 35B35, 35B40, 37L15, 37L7
Mots clés: onde progressive, stabilité globale, équation hyperbolique amortie, fonction de Lyapunov

@article{ASENS_2009_4_42_1_103_0,
     author = {Gallay, Thierry and Joly, Romain},
     title = {Global stability of travelling fronts for a damped wave equation with bistable nonlinearity},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {42},
     year = {2009},
     pages = {103-140},
     doi = {10.24033/asens.2091},
     mrnumber = {2518894},
     zbl = {1169.35041},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2009_4_42_1_103_0}
}

Gallay, Thierry; Joly, Romain. Global stability of travelling fronts for a damped wave equation with bistable nonlinearity. Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009) pp. 103-140. doi : 10.24033/asens.2091. http://gdmltest.u-ga.fr/item/ASENS_2009_4_42_1_103_0/

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