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

Thomas Duyckaerts; Carlos E. Kenig; Frank Merle

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 3, page 533-599
- ISSN: 1435-9855

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topDuyckaerts, Thomas, Kenig, Carlos E., and Merle, Frank. "Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation." Journal of the European Mathematical Society 013.3 (2011): 533-599. <http://eudml.org/doc/277181>.

@article{Duyckaerts2011,

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.},

author = {Duyckaerts, Thomas, Kenig, Carlos E., Merle, Frank},

journal = {Journal of the European Mathematical Society},

keywords = {nonlinear wave equation; type II blow-up; blow-up profile; blow-up profile; nonlinear wave equation; type II blow-up},

language = {eng},

number = {3},

pages = {533-599},

publisher = {European Mathematical Society Publishing House},

title = {Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation},

url = {http://eudml.org/doc/277181},

volume = {013},

year = {2011},

}

TY - JOUR

AU - Duyckaerts, Thomas

AU - Kenig, Carlos E.

AU - Merle, Frank

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

JO - Journal of the European Mathematical Society

PY - 2011

PB - European Mathematical Society Publishing House

VL - 013

IS - 3

SP - 533

EP - 599

AB - 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.

LA - eng

KW - nonlinear wave equation; type II blow-up; blow-up profile; blow-up profile; nonlinear wave equation; type II blow-up

UR - http://eudml.org/doc/277181

ER -

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