LONG TIME SOLUTIONS FOR QUASI-LINEAR HAMILTONIAN PERTURBATIONS OF SCHRÖDINGER AND KLEIN-GORDON EQUATIONS ON TORI - Laboratoire de Mathématiques Jean Leray Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2020

LONG TIME SOLUTIONS FOR QUASI-LINEAR HAMILTONIAN PERTURBATIONS OF SCHRÖDINGER AND KLEIN-GORDON EQUATIONS ON TORI

Résumé

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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Dates et versions

hal-02940722 , version 1 (16-09-2020)

Identifiants

  • HAL Id : hal-02940722 , version 1

Citer

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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