# Threshold for blowup for the supercritical cubic wave equation

## TL;DR

This paper numerically investigates the threshold for blowup in the supercritical cubic wave equation, analyzing stability and spectral properties of self-similar solutions in dimensions five and seven.

## Contribution

It provides numerical evidence for the blowup threshold and explores the spectral stability of self-similar solutions in supercritical dimensions.

## Key findings

- Self-similar blowup solution is at the threshold between blowup and dispersion in dimensions 5 and 7.
- Spectral analysis supports the stability properties of the blowup solution.
- Numerical computations of the spectrum in various supercritical dimensions were performed.

## Abstract

We consider the focusing cubic wave equation in the energy supercritical case, i.e., in dimensions $d \geq 5$. 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.