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

By Thomas Duyckaerts, Carlos E. Kenig, F. Merle. 2009

184 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/32f8ba649e99fe3b87f61d370538e12e2f2c8774

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Raphael Cote
2. B. Dodson
3. Thomas Duyckaerts
4. Carlos E. Kenig
5. A. Lawrie
6. F. Merle
7. Hao Jia
8. Aynur Bulut

In Control and Systems Engineering

1. Raphael Cote
2. B. Dodson
3. Carlos E. Kenig
4. A. Lawrie
5. F. Merle
6. Thomas Duyckaerts
7. Hao Jia
8. Pierre Raphael

In Mathematical Physics

1. Raphael Cote
2. B. Dodson
3. A. Lawrie
4. Carlos E. Kenig
5. Thomas Duyckaerts
6. F. Merle
7. Pierre Raphael
8. Hao Jia

Citations in the corpus

• Scattering for the radial 3D cubic wave equation (2014, background): “The subsequent work of Duyckaerts, Kenig, and Merle [10, 11, 12, 13] resulted in a classification of possible dynamics for large energies.”
• Soliton resolution for equivariant wave maps to the sphere (2013, methodology): “We now do an induction on the energy in the spirit of the Kenig Merle concentration compactness argument [15, 16].”
• Asymptotic decomposition for semilinear Wave and equivariant wave map equations (2015, background): “The works [13, 15] introduced a very natural and powerful tool, called “channels of energy”, which among many other things, implies that if a solution is not a soliton, then it will radiate energy to large distances, thus proving the crucial “rigidity theorem” for compact solutions.”
• Classification of radial solutions of the focusing, energy-critical wave equation (2012, methodology): “2 generalize Theorem 2 of [14] (for the case N = 3), which states that any radial solution of (1.”