Minoration du temps d’existence pour l’équation de Klein-Gordon non-linéaire en dimension 1 d’espace

Abstract

Let u be a solution to the semilinear Klein-Gordon equation in one space dimension ∂2tu−∂2xu+u=F(u,∂tu,∂xu) where F is a quadratic nonlinearity and the Cauchy data u|t=0 = ϵu0, ∂tu|t=0 = ϵu1 are small in C0∞. Moriyama, Tonegawa and Tsutsumi have shown that the time of existence of the solution satisfies lim infϵ→0ϵ2 log Tϵ ≥ 0. The aim of this paper is to prove that lim infϵ→0ϵ2 log Tϵ ≥ A for a constant A which can be explicitly computed from the Cauchy data and the nonlinearity.