Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres

Abstract

78 pagesThe Hamiltonian \int_X(\abs{\partial_t u}^2 + \abs{\nabla u}^2 + \m^2\abs{u}^2)\,dx, defined on functions on $\R\times X$, where $X$ is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. We consider perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of $u$. The associated PDE is then a quasi-linear Klein-Gordon equation. We show that, when $X$ is the sphere, and when the mass parameter \m is outside an exceptional subset of zero measure, smooth Cauchy data of small size $\epsilon$ give rise to almost global solutions, i.e. solutions defined on a time interval of length $c_N\epsilon^{-N}$ for any $N$. Previous results were limited either to the semi-linear case (when the perturbation of the Hamiltonian depends only on $u$) or to the one dimensional problem. The proof is based on a quasi-linear version of the Birkhoff normal forms method, relying on convenient generalizations of para-differential calculus