Global dynamics above the ground state energy for the one-dimensional NLKG equation
Abstract
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-1}u, 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.
Keywords
Cite
@article{arxiv.1011.1776,
title = {Global dynamics above the ground state energy for the one-dimensional NLKG equation},
author = {Joachim Krieger and Kenji Nakanishi and Wilhelm Schlag},
journal= {arXiv preprint arXiv:1011.1776},
year = {2010}
}