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Annales scientifiques de l'ENS - Parutions - série 4, 50 (2017)

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ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 50, fascicule 6 (2017)

Nicolas Burq, Geneviève Raugel, Wilhelm Schlag
Long time dynamics for damped Klein-Gordon equations
Annales scientifiques de l'ENS 50, fascicule 6 (2017), 1447-1498

Télécharger cet article : Fichier PDF

Résumé :
Dynamique en temps grand des solutions de l'équation de Klein-Gordon amortie
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 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).

Mots-clefs : Équation de Klein-Gordon amortie, non-linéarité sous-critique focalisante, solutions radiales, convergence, variétés invariantes, variétés centrales, condition d'Ambrosetti-Rabinowitz, estimations de Strichartz.

Abstract:
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^1L^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).

Keywords: Klein-Gordon equation with dissipation, subcritical focusing nonlinearity, radial solutions, convergence, invariant manifolds, center manifolds, Ambrosetti-Rabinowitz condition, Strichartz estimates.

Class. math. : 35B.., 35B40, 35L05, 35L71, 37L10, 37L50, 37L45.


ISSN : 0012-9593
DOI : doi:10.24033/asens.2349
Publié avec le concours de : Centre National de la Recherche Scientifique

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