Long time solutions for quasi-linear Hamiltonian perturbations of Schr\"odinger and Klein-Gordon equations on tori

TL;DR

This paper proves that solutions to certain quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori have significantly longer lifespans than previously known, especially for small initial data.

Contribution

It establishes improved lower bounds on the lifespan of solutions for these equations, extending results to quasi-linear cases and higher dimensions with novel techniques.

Findings

Lifespan for Schrödinger equation is at least order ε^{-4}

Lifespan for Klein-Gordon equation is at least order ε^{-8/3} in dimensions d≥3

Lifespan for Klein-Gordon with semi-linear perturbations is at least order ε^{-10/3} for d≥4

Abstract

We consider quasi-linear, Hamiltonian perturbations of the cubic Schr\"odinger and of the cubic (derivative) Klein-Gordon equations on the $d$ dimensional torus. If $\varepsilon\ll1$ is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time $\varepsilon^{-2}$. More precisely, concerning the Schr\"odinger equation we show that the lifespan is at least of order $O(\varepsilon^{-4})$, in the Klein-Gordon case, we prove that the solutions exist at least for a time of order $O(\varepsilon^{-{8/3}^{-}})$ as soon as $d\geq3$. 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 $O(\varepsilon^{-{10/3}^-})$, improving, for cubic non-linearities and $d\geq4$, the general results in [17,24].