Vol. 4, No. 3, 2011

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Scattering threshold for the focusing nonlinear Klein–Gordon equation

Slim Ibrahim, Nader Masmoudi and Kenji Nakanishi

Vol. 4 (2011), No. 3, 405–460

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.

nonlinear Klein–Gordon equation, scattering theory, blow-up solution, ground state, Sobolev critical exponent, Trudinger–Moser inequality
Mathematical Subject Classification 2000
Primary: 35L70, 35B40, 35B44, 47J30
Received: 28 January 2010
Revised: 11 May 2010
Accepted: 8 June 2010
Published: 28 December 2011
Slim Ibrahim
Department of Mathematics and Statistics
University of Victoria
PO Box 3060 STN CSC
Victoria V8P 5C3
Nader Masmoudi
Courant Institute for Mathematical Sciences
New York University
New York, NY 10012-1185
United States
Kenji Nakanishi
Department of Mathematics
Kyoto University
Kyoto 606-8502