Geodesic
Long time dynamics for damped Klein-Gordon equations
[Dynamique en temps grand des solutions de l'équation de Klein-Gordon amortie]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1447-1498.

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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 H1×L2. In particular, any global in positive times solution is bounded in positive times. The result applies to standard energy subcritical focusing nonlinearities |u|p-1u, 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).

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 H1×L2. 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-1u, 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).

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.
Mots-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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     title = {Long time dynamics for damped {Klein-Gordon} equations},
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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://geodesic.mathdoc.fr/articles/10.24033/asens.2349/

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