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

Gallay, Thierry; Joly, Romain

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 1, pp. 103-140.

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

Classification : 35B35, 35B40, 37L15, 37L7
Keywords: travelling front, global stability, damped wave equation, Lyapunov function
Mot clés : onde progressive, stabilité globale, équation hyperbolique amortie, fonction de Lyapunov

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