Global stability of travelling fronts for a damped wave equation with bistable nonlinearity
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 1, p. 103-140

We consider the damped wave equation αu tt +u t =u xx -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-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 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.

Nous étudions l’équation hyperbolique amortie αu tt +u t =u xx -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-st) 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 vers une onde progressive lorsque t+. 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.

DOI : https://doi.org/10.24033/asens.2091
Classification:  35B35,  35B40,  37L15,  37L7
Keywords: travelling front, global stability, damped wave equation, Lyapunov function
@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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {1},
     year = {2009},
     pages = {103-140},
     doi = {10.24033/asens.2091},
     zbl = {1169.35041},
     mrnumber = {2518894},
     language = {en},
     url = {http://www.numdam.org/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, Serie 4, Volume 42 (2009) no. 1, pp. 103-140. doi : 10.24033/asens.2091. http://www.numdam.org/item/ASENS_2009_4_42_1_103_0/

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