Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation
For researchers in nonlinear PDEs, this provides numerical evidence of intricate dynamics near the threshold, though findings are preliminary and lack definitive conclusions.
The paper numerically investigates the boundary between blowup and global existence for the focusing cubic nonlinear Klein-Gordon equation in 3D with radial data, revealing complex fine-scale structures beyond simple smooth manifold expectations.
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.