Co-dimension one stable blowup for the supercritical cubic wave equation

TL;DR

This paper constructs an explicit self-similar blowup solution for the supercritical cubic wave equation in dimensions five and higher, and proves its stability in dimension seven without symmetry assumptions, revealing a co-dimension one manifold of initial data leading to blowup.

Contribution

It introduces a new explicit self-similar blowup solution for supercritical cubic wave equations and establishes its stability properties in dimension seven without symmetry constraints.

Findings

Existence of a self-similar blowup solution in all supercritical dimensions $d \,\geq\, 5$.

Proof of stability of this solution in dimension 7 without symmetry assumptions.

Identification of a co-dimension one Lipschitz manifold of initial data leading to blowup.

Abstract

For the focusing cubic wave equation, we find an explicit, non-trivial self-similar blowup solution $u^*_T$, which is defined on the whole space and exists in all supercritical dimensions $d \geq 5$. For $d=7$, we analyze its stability properties without any symmetry assumptions and prove the existence of a set of perturbations which lead to blowup via $u^*_T$ in a backward light cone. Moreover, this set corresponds to a co-dimension one Lipschitz manifold modulo translation symmetries in similarity coordinates.