Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case

TL;DR

This paper extends the understanding of type II blow-up solutions for the energy-critical wave equation to non-radial cases, showing they asymptotically resemble a rescaled stationary solution combined with a Lorentz transform.

Contribution

It generalizes previous radial results to non-radial solutions, demonstrating their universal blow-up profile under smallness conditions.

Findings

Type II blow-up solutions asymptotically resemble a stationary solution plus a Lorentz transform.

The universality of the blow-up profile is established beyond radial symmetry.

Smallness assumptions are crucial for the asymptotic description.

Abstract

Following our previous paper in the radial case, we consider blow-up type II solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of W concentrating at the origin.