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

Abstract

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 .