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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 ε 1 is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time ε −2. More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order O(ε −4) while in the Klein-Gordon case, we prove that the solutions exist at least for a time of order O(ε −8/3 +) as soon as d ≥ 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(ε −10/3 +), improving, for cubic non-linearities and d ≥ 4, the general result in [17], where the time of existence is decreasing with respect to the dimension.
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https://hal.archives-ouvertes.fr/hal-02940722
Contributor : Felice Iandoli <>
Submitted on : Wednesday, September 16, 2020 - 2:52:28 PM
Last modification on : Saturday, September 19, 2020 - 4:26:35 AM

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  • HAL Id : hal-02940722, version 1

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Roberto Feola, Benoît Grébert, Felice Iandoli. LONG TIME SOLUTIONS FOR QUASI-LINEAR HAMILTONIAN PERTURBATIONS OF SCHRÖDINGER AND KLEIN-GORDON EQUATIONS ON TORI. 2020. ⟨hal-02940722⟩

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