Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation

Abstract

We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation utt − uxx + u − |u|2αu = 0 on the line. Using mixed numerical and analytical methods we find that solutions starting from even initial data, fine-tuned to the threshold, are trapped by the static solution S for intermediate times. The details of trapping are shown to depend on the power α, namely, we observe fast convergence to S for α > 1, slow convergence for α = 1, and very slow (if any) convergence for 0 < α < 1. Our findings are complementary with respect to the recent rigorous analysis of the same problem (for α > 2) by Krieger, Nakanishi, and Schlag [“Global dynamics above from the ground state energy for the one-dimensional NLKG equation,” preprint arXiv:1011.1776 [math.AP]].