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Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it scatters. The proof relies on the space-time resonance approach; it turns out that the resonant structure of this equation has features which were not studied before, but which are generic in some sense.

This article is a short exposition of the space-time resonances method. It was introduced by Masmoudi, Shatah, and the author, in order to understand global existence for nonlinear dispersive equations, set in the whole space, and with small data. The idea is to combine the classical concept of resonances, with the feature of dispersive equations: wave packets propagate at a group velocity which depends on their frequency localization. The analytical method which follows from this idea turns out...

Nous considérons dans cet article l’équation des ondes semilinéaire critique $${\left(NLW\right)}_{2^{*}-1}\left\{\begin{array}{c}\square u+{\left|u\right|}^{2^{*}-2}u=0\hfill \\ u_{|t=0}=u_0\hfill \\ {\partial }_t{u}_{|t=0}=u_1,\hfill \end{array}\right.$$ posée dans tout l’espace ${ℝ}^d$, avec $2^{*}=\frac{2d}{d-2}·$ Shatah et Struwe [31] ont prouvé que si les données initiales sont d’énergie finie, c’est à dire si $(u_0,u_1)\in {\dot{H}}^1\times L^2$, alors il existe une solution globale. Planchon [22] a montré que c’est aussi le cas pour certaines données initiales d’énergie infinie : il suffit que les données initiales soient de norme petite dans ${\dot{B}}_{2,\infty }^1\times {\dot{B}}_{2,\infty }^0$. Nous construisons ici des solutions globales de ${\left(NLW\right)}_{2^{*}-1}$ pour...