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

### Carlos E. Kenig

University of Chicago, USA### Thomas Duyckaerts

Institut Galilée, Université Paris 13, Villetaneuse, France### Frank Merle

Université de Cergy-Pontoise, France

## 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.

## Cite this article

Carlos E. Kenig, Thomas Duyckaerts, Frank Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation. J. Eur. Math. Soc. 13 (2011), no. 3, pp. 533–599

DOI 10.4171/JEMS/261