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15 March 2009 Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation
Joachim Krieger, Wilhelm Schlag, Daniel Tataru
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Duke Math. J. 147(1): 1-53 (15 March 2009). DOI: 10.1215/00127094-2009-005

Abstract

Given ν>1/2 and δ>0 arbitrary, we prove the existence of energy solutions of ttuΔuu5=0 (0.1) in R3+1 which blow up exactly at r=t=0 as t0. These solutions are radial and of the form u=λ(t)1/2W(λ(t)r)+η(r,t) inside the cone rt, where λ(t)=t1ν, W(r)=(1+r2/3)1/2 is the stationary solution of (0.1), and η is a radiation term with [rt](|η(x,t)|2+|ηt(x,t)|2+|η(x,t)|6)dx0, t0. Outside of the light-cone, there is the energy bound [r>t](|u(x,t)|2+|ut(x,t)|2+|u(x,t)|6)dx<δ for all small t>0. The regularity of u increases with ν. As in our accompanying article on wave maps [10], the argument is based on a renormalization method for the “soliton profile” W(r)

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Joachim Krieger. Wilhelm Schlag. Daniel Tataru. "Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation." Duke Math. J. 147 (1) 1 - 53, 15 March 2009. https://doi.org/10.1215/00127094-2009-005

Information

Published: 15 March 2009
First available in Project Euclid: 26 February 2009

zbMATH: 1170.35066
MathSciNet: MR2494455
Digital Object Identifier: 10.1215/00127094-2009-005

Subjects:
Primary: 35L75
Secondary: 58J50

Rights: Copyright © 2009 Duke University Press

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Vol.147 • No. 1 • 15 March 2009
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