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arXiv · 2009.07553

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

Roberto Feola·Benoît Grébert·Felice Iandoli
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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 $\varepsilon\ll1$ 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 Schrödinger 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].

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Roberto Feola, Benoît Grébert, Felice Iandoli. 2022-02-14. Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori. https://doi.org/10.2140/apde.2023.16.1133

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