On blowup for the supercritical quadratic wave equation

TL;DR

This paper investigates singularity formation in the supercritical quadratic wave equation, introducing a new self-similar blowup solution, analyzing its stability, and establishing regularity persistence in similarity coordinates.

Contribution

It presents a new explicit self-similar blowup solution for all dimensions $d \,\geq\, 7$, and analyzes its stability and regularity properties.

Findings

Existence of a new self-similar blowup solution for all $d \geq 7$

Stability analysis of the blowup solution in dimensions 7 and 9

Persistence of regularity in similarity coordinates

Abstract

We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d \geq 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^*$ which exists for all $d \geq 7$. For $d=9$, we study the stability of $u^*$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^*$. In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d=7$ and $d=9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.