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Scattering threshold
for the focusing nonlinear Klein–Gordon equation
Slim Ibrahim, Nader Masmoudi and Kenji Nakanishi |
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Vol. 4 (2011), No. 3, 405–460
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Abstract |
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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 critical wave and Schrödinger equations. Our result includes the 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. |
Keywords
nonlinear Klein–Gordon equation, scattering theory, blow-up
solution, ground state, Sobolev critical exponent,
Trudinger–Moser inequality
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Mathematical Subject Classification 2000
Primary: 35L70, 35B40, 35B44, 47J30
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Milestones
Received: 28 January 2010
Revised: 11 May 2010
Accepted: 8 June 2010
Published: 28 December 2011
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Authors |
| Slim Ibrahim | |
| Department of Mathematics and
Statistics University of Victoria PO Box 3060 STN CSC Victoria V8P 5C3 Canada |
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| http://www.math.uvic.ca/~ibrahim/ | |
| Nader Masmoudi | |
| Courant Institute for Mathematical
Sciences New York University New York, NY 10012-1185 United States |
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| http://www.math.nyu.edu/faculty/masmoudi | |
| Kenji Nakanishi | |
| Department of Mathematics Kyoto University Kyoto 606-8502 Japan |
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