Partial Differential Equations
Long time existence problems for semilinear Klein–Gordon equations
Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 149-154.

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,tu,u) which are quadratic in (tu,u) and do not have any other special structure. We prove that the solution exists over an interval of time exponential in ε2/3, where ε is the size in Hs 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,tu,u) quadratiques en (tu,u), et ne vérifiant aucune autre condition de structure particulière, en dimension grande d4. Nous montrons que le problème considéré admet des solutions définies sur un intervalle de temps exponentiel en ε2/3, où ε désigne la taille dans Hs des données de Cauchy.

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DOI: 10.1016/j.crma.2007.11.012
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. http://www.numdam.org/articles/10.1016/j.crma.2007.11.012/

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