Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation

TL;DR

This paper uses numerical methods to explore the boundary between blowup and global existence for the focusing cubic nonlinear Klein-Gordon equation in three dimensions, revealing complex structures near the scattering threshold.

Contribution

It provides new numerical insights into the blowup/global existence boundary, highlighting intricate structures and extending previous theoretical work.

Findings

Revealed complex structures near the scattering boundary.

Numerical evidence suggests the boundary may not be a smooth manifold.

Extended analysis of the blowup vs. global existence dichotomy.

Abstract

We present some numerical findings concerning the nature of the blowup vs. global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation in three dimensions for radial data. The context of this study is provided by the classical paper by Payne, Sattinger from 1975, as well as the recent work by K. Nakanishi, and the second author arXiv:1005.4894. Specifically, we numerically investigate the boundary of the forward scattering region. At this point we do not have sufficient numerical evidence that might indicate whether or not the boundary remains a smooth manifold for general energies. In this updated version we include some fine-scale computations that reveal more complicated structures than one might expect.