Threshold for blowup for the supercritical cubic wave equation

Abstract

We consider the focusing cubic wave equation in the energy supercritical case, i.e., in dimensions $d\ge 5$. For this model an explicit nontrivial self-similar blowup solution was recently found by the first and third author in \cite{GlogicSchoerkhuber}. Furthermore, the solution is proven to be co-dimension one stable in $d=7$. In this paper, we study the equation from a numerical point of view. For $d=5$ and $d=7$ in the radial case, we provide evidence that this solution is at the threshold between generic ODE blowup and dispersion. In addition, we investigate the spectral problem that underlies the stability analysis and compute the spectrum in general supercritical dimensions.

19 pages, 6 figures, 2 tables; v2 introduction restructured, sections 2 and 3 expanded, accepted for publication in Nonlinearity