Co-dimension one stable blowup for the quadratic wave equation beyond the light cone

TL;DR

This paper proves the nonlinear stability of a specific self-similar blowup solution to the quadratic wave equation in seven dimensions, using hyperboloidal similarity coordinates and spectral analysis, revealing exactly one instability mode.

Contribution

It introduces a new canonical method employing hyperboloidal similarity coordinates to analyze unstable self-similar solutions in nonlinear wave equations.

Findings

Proves stability near the Cauchy horizon for the blowup solution.

Identifies exactly one genuine spectral instability.

Develops a new approach for studying unstable self-similar solutions.

Abstract

We study the stability of an explicitly known, non-trivial self-similar blowup solution of the quadratic wave equation in the lowest energy supercritical dimension $d = 7$. This solution blows up at a single point and extends naturally away from the singularity. By using hyperboloidal similarity coordinates, we prove the conditional nonlinear asymptotic stability of this solution under small, compactly supported radial perturbations in a region of spacetime which can be made arbitrarily close to the Cauchy horizon of the singularity. To achieve this, we rigorously solve the underlying spectral problem and show that the solution has exactly one genuine instability. The unstable nature of the solution requires a careful construction of suitably adjusted initial data at $t = 0$, which, when propagated to a family of spacelike hypersurfaces of constant hyperboloidal time, takes the required form to guarantee convergence. By this, we introduce a new canonical method to investigate unstable self-similar solutions for nonlinear wave equations within the framework of hyperboloidal similarity coordinates.