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Long time solutions
for quasilinear Hamiltonian perturbations of Schrödinger and
Klein–Gordon equations on tori
Roberto Feola, Benoît Grébert and Felice Iandoli |
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Vol. 16 (2023), No. 5, 1133–1203
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Abstract |
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We consider quasilinear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein–Gordon equations on the -dimensional torus. If is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time . More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order , and in the Klein–Gordon case we prove that the solutions exist at least for a time of order as soon as . Regarding the Klein–Gordon equation, our result presents novelties also in the case of semilinear perturbations: we show that the lifespan is at least of order , improving, for cubic nonlinearities and , the general results of Delort (J. Anal. Math. 107 (2009), 161–194) and Fang and Zhang (J. Differential Equations 249:1 (2010), 151–179). |
Keywords
quasilinear equations, paradifferential calculus, energy
estimates, small divisors
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Mathematical Subject Classification
Primary: 37K45, 35S50, 35B35, 35B45, 35L05, 35Q55
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Milestones
Received: 5 October 2020
Revised: 14 September 2021
Accepted: 7 January 2022
Published: 12 August 2023
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Authors |
| Roberto Feola | |
| Dipartimento di Matematica e
Fisica Universitá degli studi Roma Tre Rome Italy |
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| Benoît Grébert | |
| Laboratoire de Mathématiques Jean
Leray Université de Nantes, UMR CNRS 6629 Nantes France |
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| Felice Iandoli | |
| Dipartimento di Matematica e
Informatica Università della Calabria Rende Italy |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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