Existence en temps grand pour des équations de Klein–Gordon à petite donnée initiale sur une structure de Toeplitz

Abstract

We prove a long time existence, of order C(r)ε−r for all r⩾3, for small solutions, of order ε≪1, in high Sobolev norms of Klein–Gordon equation with Hamiltonian nonlinearities. Manifolds studied are endowed with Toeplitz structures in the sense of Boutet de Monvel and Guillemin. We also make a geometric assumption about periodicity of the Toeplitz pseudo-differential operator Hamiltonian flow. This ensures a useful spectral localization. Our approach follows Delort and Szeftelʼs and Bambusi, Delort, Grebert and Szeftelʼs works on the spheres and Zoll manifolds and uses Birkhoff normal forms at any order. The context of Toeplitz structures allows us to generalize all the previous cases (torus, spheres and Zoll manifolds) and to deal with new linearities involving Szegö projectors.