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.

We consider the damped wave equation $\alpha {u}_{tt}+{u}_{t}={u}_{xx}-{V}^{\text{'}}\left(u\right)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u\left(x,t\right)=h\left(x-st\right)$ 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.

Nous étudions l’équation hyperbolique amortie $\alpha {u}_{tt}+{u}_{t}={u}_{xx}-{V}^{\text{'}}\left(u\right)$ sur la droite réelle, où $V$ est un potentiel bistable. Cette équation possède des ondes progressives de la forme $u\left(x,t\right)=h\left(x-st\right)$ 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.

DOI: 10.24033/asens.2091
Classification: 35B35,  35B40,  37L15,  37L7
Keywords: travelling front, global stability, damped wave equation, Lyapunov function
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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://archive.numdam.org/articles/10.24033/asens.2091/

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