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1 November 2009 Scattering for the two-dimensional energy-critical wave equation
Slim Ibrahim, Mohamed Majdoub, Nader Masmoudi, Kenji Nakanishi
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Duke Math. J. 150(2): 287-329 (1 November 2009). DOI: 10.1215/00127094-2009-053

Abstract

We investigate existence and asymptotic completeness of the wave operators for the nonlinear Klein-Gordon equation with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser-type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. An interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities cannot control the nonlinear term uniformly on each time interval: it crucially depends on how much the energy is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and only after that we can apply Bourgain's induction argument (or any other similar one)

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Slim Ibrahim. Mohamed Majdoub. Nader Masmoudi. Kenji Nakanishi. "Scattering for the two-dimensional energy-critical wave equation." Duke Math. J. 150 (2) 287 - 329, 1 November 2009. https://doi.org/10.1215/00127094-2009-053

Information

Published: 1 November 2009
First available in Project Euclid: 16 October 2009

zbMATH: 1206.35175
MathSciNet: MR2569615
Digital Object Identifier: 10.1215/00127094-2009-053

Subjects:
Primary: 35L70
Secondary: 35B33 , 35B40 , 35Q55 , 37K05 , 37L50

Rights: Copyright © 2009 Duke University Press

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Vol.150 • No. 2 • 1 November 2009
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