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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
H 1
critical wave and Schrödinger equations. Our result includes the
H 1
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
Mathematical Subject Classification 2000
Primary: 35L70, 35B40, 35B44, 47J30
Milestones
Received: 28 January 2010
Revised: 11 May 2010
Accepted: 8 June 2010
Published: 28 December 2011