Analysis & PDE

  • Anal. PDE
  • Volume 4, Number 3 (2011), 405-460.

Scattering threshold for the focusing nonlinear Klein–Gordon equation

Slim Ibrahim, Nader Masmoudi, and Kenji Nakanishi

Full-text: Open access

Abstract

We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein–Gordon equation, in the spirit of Kenig and Merle for the H1 critical wave and Schrödinger equations. Our result includes the H1 critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger–Moser inequality. The main difficulty is the lack of scaling invariance in both the linear and the nonlinear terms.

Article information

Source
Anal. PDE, Volume 4, Number 3 (2011), 405-460.

Dates
Received: 28 January 2010
Revised: 11 May 2010
Accepted: 8 June 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731147

Digital Object Identifier
doi:10.2140/apde.2011.4.405

Mathematical Reviews number (MathSciNet)
MR2872122

Zentralblatt MATH identifier
1270.35132

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations 35B40: Asymptotic behavior of solutions 35B44: Blow-up 47J30: Variational methods [See also 58Exx]

Keywords
nonlinear Klein–Gordon equation scattering theory blow-up solution ground state Sobolev critical exponent Trudinger–Moser inequality

Citation

Ibrahim, Slim; Masmoudi, Nader; Nakanishi, Kenji. Scattering threshold for the focusing nonlinear Klein–Gordon equation. Anal. PDE 4 (2011), no. 3, 405--460. doi:10.2140/apde.2011.4.405. https://projecteuclid.org/euclid.apde/1513731147


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