Long time existence problems for semilinear Klein–Gordon equations

Benoaga, Laurentiu

doi:10.1016/j.crma.2007.11.012

Partial Differential Equations

Long time existence problems for semilinear Klein–Gordon equations

Presented by: Gilles Lebeau

¹Université Paris 13, Institut Galilée, département de mathématiques, 99, avenue J.-B. Clément, 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 149-154

Abstract (Original language)
Résumé

We study a problem of almost global existence for solutions of semilinear Klein–Gordon equations with small weakly decaying Cauchy data. Our work concerns nonlinearities $P (u, \partial_{t} u, \nabla u)$ which are quadratic in $(\partial_{t} u, \nabla u)$ and do not have any other special structure. We prove that the solution exists over an interval of time exponential in $ε^{- 2 / 3}$ , where ε is the size in $H^{s}$ of the Cauchy data. The main difficulty is to construct, using suitable local cut-offs, the function spaces in which the nonlinearities verify the necessary estimates for the proof of a contraction property.

Notre travail est consacré à un problème d'existence presque globale pour des solutions d'équations de Klein–Gordon semi-linéaire à données petites faiblement décroissantes. Nous abordons le cas de non-linéarités $P (u, \partial_{t} u, \nabla u)$ quadratiques en $(\partial_{t} u, \nabla u)$ , et ne vérifiant aucune autre condition de structure particulière, en dimension grande $d ⩾ 4$ . Nous montrons que le problème considéré admet des solutions définies sur un intervalle de temps exponentiel en $ε^{- 2 / 3}$ , où ε désigne la taille dans $H^{s}$ des données de Cauchy.

Received: 2007-05-02
Accepted: 2007-11-20
Published online: 2008-01-11

DOI: 10.1016/j.crma.2007.11.012

Author's affiliations:

Benoaga, Laurentiu ¹

¹ Université Paris 13, Institut Galilée, département de mathématiques, 99, avenue J.-B. Clément, 93430 Villetaneuse, France

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Benoaga, Laurentiu. Long time existence problems for semilinear Klein–Gordon equations. Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 149-154. doi: 10.1016/j.crma.2007.11.012

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