Temps d'existence pour l'équation de Klein-Gordon semi-linéaire à données petites périodiques

We study lower bounds for the maximal time of existence T_∈ of a smooth solution to a semi-linear Klein-Gordon equation □u + u = F(u, u'), with periodic Cauchy data of small size ∈. If F vanishes at order r at 0, we prove that T_∈ ≥ c∈^-2 if r = 2, T∈ ≥ c∈^-(r-1)| log _∈|^-(r-3) if r ≥ 3. We construct examples showing the optimality of these results for convenient values of r.