Global dynamics above the ground state energy for the one-dimensional NLKG equation

In this paper we obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the line with focusing nonlinearity , provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case (Nakanishi and Schlag in Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, preprint, 2010), the major difference being in the construction of the center-stable manifold. The difficulty there lies with the weak dispersive decay of 1-dimensional NLKG. In order to address this specific issue, we establish local dispersive estimates for the perturbed linear Klein-Gordon equation, similar to those of Mizumachi (J Math Kyoto Univ 48(3):471-497, 2008). The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.