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Comptes Rendus Mathématique
Volume 346, n° 3-4
pages 149-154 (février 2008)
Doi : 10.1016/j.crma.2007.11.012
Received : 2 May 2007 ;  accepted : 20 November 2007
Long time existence problems for semilinear Klein–Gordon equations
Problèmes d’existence en temps grand pour des équations de Klein–Gordon non-linéaires
 

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

Abstract

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   which are quadratic in   and do not have any other special structure . We prove that the solution exists over an interval of time exponential in  , where is the size in   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. To cite this article: L. Benoaga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.
Résumé

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   quadratiques en  , et ne vérifiant aucune autre condition de structure particulière , en dimension grande  . Nous montrons que le problème considéré admet des solutions définies sur un intervalle de temps exponentiel en  , où désigne la taille dans   des données de Cauchy. Pour citer cet article : L. Benoaga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.


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