Benoît Grébert (Université de Nantes)
I will present an almost global existence result for semilinear Hamiltonian PDEs on compact boundaryless manifolds. As a main application, we prove the almost global existence of small solutions to nonlinear Klein–Gordon equations on such manifolds: for almost all mass, any arbitrarily large r and sufficiently large s, solutions with initial data of sufficiently small size ε ≪ 1 in the Sobolev space Hs ×Hs−1 exist and remain in Hs ×Hs−1 for polynomial times |t| ≤ ε−r. Surprisingly, it turns out that the geometry of the manifold has no influence. The abstract result applies to equations satisfying very weak non resonance conditions and natural multilinear estimates. (Joint work with D. Bambusi, J. Bernier and R. Imekraz)