Delort, J.-M. Time of existence for the semilinear Klein-Gordon equation with periodic small data. (Temps d’existence pour l’équation de Klein-Gordon semi-linéaire à données petites périodiques.) (French) Zbl 0902.35108 Am. J. Math. 120, No. 3, 663-689 (1998). Summary: We study lower bounds for the maximal time of existence \(T_\varepsilon\) of a smooth solution to a semilinear Klein-Gordon equation \(\square u+u= F(u,u')\), with periodic Cauchy data of small size \(\varepsilon\). If \(F\) vanishes at order \(r\) at \(0\), we prove that \[ T_\varepsilon\geq c\varepsilon^{-2}\quad\text{if} \quad r=2,\quad T_\varepsilon\geq c\varepsilon^{-(r- 1)}|\log\varepsilon|^{- (r-3)}\quad\text{if }r\geq 3. \] We construct examples showing the optimality of these results for convenient values of \(r\). Cited in 14 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:maximal time of existence; semilinear Klein-Gordon equation PDFBibTeX XMLCite \textit{J. M. Delort}, Am. J. Math. 120, No. 3, 663--689 (1998; Zbl 0902.35108) Full Text: DOI Link