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

We present some numerical findings concerning the nature of the blowup versus 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 classic paper by Payne and Sattinger (1975 Israel J. Math. 22 273–303), as well as the recent work by Nakanishi and Schlag (2010 J. Diff. Eqns arXiv:1005.4894). Specifically, we numerically investigate the boundary of the forward scattering region. While the results of (2010 J. Diff. Eqns arXiv:1005.4894) guarantee that this boundary is smooth at energies which are near the ground state energy, it is currently unknown whether or not it continues to be a smooth manifold at higher energies. While we do not find convincing evidence of either smoothness or singularity formation, our numerical work does indicate that at larger energies the boundary becomes much more complicated than at energies near that of the ground state.