The Hamiltonian \(\int_X(\lvert{\partial_t u}\rvert^2 + \lvert{\nabla u}\rvert^2 + \mathbf{m}^2\lvert{u}\rvert^2)\,dx\), defined on functions on \(\mathbb{R}\times X\), where \(X\) is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation.

The author considers 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. The author shows that, when \(X\) is the sphere, and when the mass parameter \(\mathbf{m}\) 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.