Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres - volume 234, number 1103

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.