Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres

The Hamiltonian integral(X) (vertical bar partial derivative(t)u vertical bar(2) + vertical bar del u vertical bar(2) + m(2) vertical bar u vertical bar(2)) dx, defined on functions on R x 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 is an element of give rise to almost global solutions, i.e. solutions defined on a time interval of length c(N)is an element of(-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.