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

TL;DR

This paper demonstrates that in three dimensions, small radial type II blow-up solutions of the energy-critical wave equation are universally characterized by a rescaled stationary solution plus a small, continuous remainder.

Contribution

It establishes the universality of blow-up profiles for small radial type II solutions, showing they are essentially rescaled stationary solutions plus a small perturbation.

Findings

Any such solution is a rescaled W plus a small remainder.

The unique radial solution compact up to scaling is W up to symmetries.

Results align with solutions constructed by Krieger, Schlag, and Tataru.

Abstract

Consider the energy critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.