  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.
