Scattering threshold for the focusing nonlinear Klein-Gordon equation

Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)405-460
Number of pages56
JournalAnalysis and PDE
Issue number3
StatePublished - 2011


  • Blow-up solution
  • Ground state
  • Nonlinear Klein-Gordon equation
  • Scattering theory
  • Sobolev critical exponent
  • Trudinger-Moser inequality

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics


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