%0 Journal Article
%T Scattering threshold for the focusing nonlinear Klein-Gordon equation
%A Slim Ibrahim
%A Nader Masmoudi
%A Kenji Nakanishi
%J Mathematics
%D 2010
%I arXiv
%X We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig-Merle for the $H^1$ critical wave and Schr\"odinger 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 lack of scaling invariance both in the linear and nonlinear terms.
%U http://arxiv.org/abs/1001.1474v3