Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori

Abstract

We consider quasi-linear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein-Gordon equations on the $d$ dimensional torus. If $\epsilon \ll 1$ is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time ${\epsilon }^{-2}$. More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order $O({\epsilon }^{-4})$, in the Klein-Gordon case, we prove that the solutions exist at least for a time of order $O({\epsilon }^{-{8/3}^{-}})$ as soon as $d\ge 3$. Regarding the Klein-Gordon equation, our result presents novelties also in the case of semi-linear perturbations: we show that the lifespan is at least of order $O({\epsilon }^{-{10/3}^{-}})$, improving, for cubic non-linearities and $d\ge 4$, the general results in [17,24].

The introduction has been expanded and some typos corrected. To appear on ANALYSIS AND PDE