2010/11/08 by Krieger, Joachim, Nakanishi, Kenji, Schlag, Wilhelm · 1 citation
#35L70 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1011.1776
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 |u|p-1u, p>5, provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case arXiv:1005.4894, 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 arXiv:math/0605031. The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.