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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
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 \(r\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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