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LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE $d$-DIMENSIONAL TORUS

Part of: Partial differential equations Equations of mathematical physics and other areas of application Hyperbolic equations and systems Infinite-dimensional Hamiltonian systems

Published online by Cambridge University Press:  06 March 2020

JOACKIM BERNIER ,
ERWAN FAOU  and
BENOÎT GRÉBERT
JOACKIM BERNIER
Affiliation:
Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, Université Paul Sabatier, F-31062Toulouse Cedex 9, France; joackim.bernier@math.univ-toulouse.fr
ERWAN FAOU
Affiliation:
Univ Rennes, INRIA, CNRS, IRMAR - UMR 6625, F-35000Rennes, France; Erwan.Faou@inria.fr
BENOÎT GRÉBERT
Affiliation:
Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, 44322Nantes Cedex 03, France; benoit.grebert@univ-nantes.fr

Abstract

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We consider the nonlinear wave equation (NLW) on the $d$-dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

MSC classification

Primary: 35L05: Wave equation 37K55: Perturbations, KAM for infinite-dimensional systems
Secondary: 35B40: Asymptotic behavior of solutions 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e12
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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