Long time dynamics for damped Klein-Gordon equations

Burq, Nicolas; Raugel, Geneviève; Schlag, Wilhelm

doi:10.24033/asens.2349

Long time dynamics for damped Klein-Gordon equations
[Dynamique en temps grand des solutions de l'équation de Klein-Gordon amortie]

Burq, Nicolas ; Raugel, Geneviève ; Schlag, Wilhelm

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1447-1498.

Résumé
Abstract

Nous démontrons que toute solution radiale d'énergie finie d'une classe générale d'équations de Klein-Gordon amorties ou bien explose en temps positif fini ou bien converge en temps positif vers une solution stationnaire dans $H^{1} \times L^{2}$ . En particulier, toute solution globale en temps positif est bornée en temps positif. Ce résultat s'applique aux non-linéarités focalisantes, sous-critiques pour l'énergie, ${| u |}^{p - 1} u$ , $1 < p < (d + 2) / (d - 2)$ , comme à toute non-linéarité, sous-critique pour l'énergie, remplissant une condition de signe de type Ambrosetti-Rabinowitz. La preuve fait appel, à la fois, à des techniques propres aux équations non linéaires dispersives et à des arguments de systèmes dynamiques (variétés invariantes dans des espaces de Banach et théorèmes de convergence).

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^{1} \times L^{2}$ . In particular, any global in positive times solution is bounded in positive times. The result applies to standard energy subcritical focusing nonlinearities ${| u |}^{p - 1} u$ , $1 < p < (d + 2) / (d - 2)$ as well as to any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).

DOI : 10.24033/asens.2349

Classification : 35B.., 35B40, 35L05, 35L71, 37L10, 37L50, 37L45.
Keywords: Klein-Gordon equation with dissipation, subcritical focusing nonlinearity, radial solutions, convergence, invariant manifolds, center manifolds, Ambrosetti-Rabinowitz condition, Strichartz estimates.
Mot clés : Équation de Klein-Gordon amortie, non-linéarité sous-critique focalisante, solutions radiales, convergence, variétés invariantes, variétés centrales, condition d'Ambrosetti-Rabinowitz, estimations de Strichartz.

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     author = {Burq, Nicolas and Raugel, Genevi\`eve and Schlag, Wilhelm},
     title = {Long time dynamics for damped {Klein-Gordon} equations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1447--1498},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {6},
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     mrnumber = {3742197},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2349/}
}

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Burq, Nicolas; Raugel, Geneviève; Schlag, Wilhelm. Long time dynamics for damped Klein-Gordon equations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1447-1498. doi : 10.24033/asens.2349. http://archive.numdam.org/articles/10.24033/asens.2349/

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