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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

Vol. 16 (2023), No. 5, 1133–1203
Abstract

We consider quasilinear, 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 𝜖2. More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order O(𝜖4), and in the Klein–Gordon case we prove that the solutions exist at least for a time of order O(𝜖83 ) as soon as d 3. 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 O(𝜖103 ), improving, for cubic nonlinearities and d 4, 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
Mathematical Subject Classification
Primary: 37K45, 35S50, 35B35, 35B45, 35L05, 35Q55
Milestones
Received: 5 October 2020
Revised: 14 September 2021
Accepted: 7 January 2022
Published: 12 August 2023
Authors
Roberto Feola
Dipartimento di Matematica e Fisica
Universitá degli studi Roma Tre
Rome
Italy
Benoît Grébert
Laboratoire de Mathématiques Jean Leray
Université de Nantes, UMR CNRS 6629
Nantes
France
Felice Iandoli
Dipartimento di Matematica e Informatica
Università della Calabria
Rende
Italy

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