Scattering for the two-dimensional energy-critical wave equation

Slim Ibrahim, Mohamed Majdoub, Nader Masmoudi, Kenji Nakanishi

Research output: Contribution to journalArticle

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).

Original languageEnglish (US)
Pages (from-to)287-329
Number of pages43
JournalDuke Mathematical Journal
Volume150
Issue number2
DOIs
StatePublished - Nov 2009

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Wave equation
Scattering
Energy
Strichartz Estimates
Nonlinear Klein-Gordon Equation
Wave Operator
Critical Case
Kinetic energy
Critical value
Completeness
Proof by induction
Trace
Infinity
Nonlinearity
Interval
Term
Estimate

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Scattering for the two-dimensional energy-critical wave equation. / Ibrahim, Slim; Majdoub, Mohamed; Masmoudi, Nader; Nakanishi, Kenji.

In: Duke Mathematical Journal, Vol. 150, No. 2, 11.2009, p. 287-329.

Research output: Contribution to journalArticle

Ibrahim, Slim ; Majdoub, Mohamed ; Masmoudi, Nader ; Nakanishi, Kenji. / Scattering for the two-dimensional energy-critical wave equation. In: Duke Mathematical Journal. 2009 ; Vol. 150, No. 2. pp. 287-329.
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