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

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.

30 images. In this updated we include results that were obtained by means of the CRAY XT5 supercomputer at the NICS, the National Institute of Computational Sciences at Oakridge Labs, Tennessee, which is part of the TeraGrid. Support by the NSF through TG-DMS110003 is gratefully acknowledged

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• Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients
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