Journal Article

Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres

J.-M. Delort and J. Szeftel

in International Mathematics Research Notices

Volume 2004, issue 37, pages 1897-1966
Published in print January 2004 | ISSN: 1073-7928
Published online January 2004 | e-ISSN: 1687-0247 | DOI: https://dx.doi.org/10.1155/S1073792804133321
Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres

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Consider a nonlinear Klein-Gordon equation on a compact manifold M with smooth Cauchy data of size ε → 0. Denote by Tε the maximal time of existence of a smooth solution. One always has Tεc εk+1 if the nonlinearity vanishes at order k at 0. We prove that when [math], k=2, and the equation is quasilinear, Tεcε−2. When [math], the equation is semilinear, and k = 2p, we show that Tεcε−2p for almost every mass. The proof of these results relies on a systematic use of normal forms methods.

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Subjects: Mathematics

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