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

@article{Duyckaerts2010UniversalityOT,
title={Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case},
author={Thomas Duyckaerts and Carlos E. Kenig and Frank Merle},
journal={arXiv: Analysis of PDEs},
year={2010}
}
• Published 2 March 2010
• Mathematics
• arXiv: Analysis of PDEs
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.
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## References

SHOWING 1-10 OF 43 REFERENCES
Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation
• Mathematics
• 2009
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
Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation
• Mathematics
• 2005
We consider finite time blow up solutions to the critical nonlinear Schrödinger equation For a suitable class of initial data in the energy space H1, we prove that the solution splits in two parts:
On universality of blow-up profile for L2 critical nonlinear Schrödinger equation
• Mathematics
• 2004
We consider finite time blow-up solutions to the critical nonlinear Schrödinger equation iut=-Δu-|u|4/Nu with initial condition u0∈H1. Existence of such solutions is known, but the complete blow-up
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation
• Mathematics
• 2006
We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static
Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation
• Mathematics
• 2002
The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term.
Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions
• Mathematics
• 2009
In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ × ℝ d with d ≥ 6. We prove the stability of solutions under the weak
Radially symmetric wave maps from (1 + 2)-dimensional Minkowski space to the sphere
Abstract. By a blow-up analysis as in [8] for a related problem we rule out concentration of energy for radially symmetric wave maps from the (1+ 2)-dimensional Minkowski space to the sphere. When
On the formation of singularities in the critical $O(3)$ $\sigma$-model
• Mathematics
• 2006
We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from ℝ 2+1 Minkowski space into the sphere S 2 . We establish rigorously and