## Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation.(English)Zbl 1230.35067

Authors’ 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 a 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 the following: 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 J. Krieger, W. Schlag and D. Tataru [Duke Math. J. 147, No. 1, 1–53 (2009; Zbl 1170.35066)]. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to $$W$$ up to symmetries.

### MSC:

 35L71 Second-order semilinear hyperbolic equations 35B44 Blow-up in context of PDEs

### Keywords:

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

Zbl 1170.35066
Full Text:

### References:

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