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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

Citations:

Zbl 1170.35066
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References:

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