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2014 Nondispersive decay for the cubic wave equation
Roland Donninger, Anil Zenginoğlu
Anal. PDE 7(2): 461-495 (2014). DOI: 10.2140/apde.2014.7.461

Abstract

We consider the hyperboloidal initial value problem for the cubic focusing wave equation

( t 2 + Δ x ) v ( t , x ) + v ( t , x ) 3 = 0 , x 3 .

Without symmetry assumptions, we prove the existence of a codimension-4 Lipschitz manifold of initial data that lead to global solutions in forward time which do not scatter to free waves. More precisely, for any δ(0,1), we construct solutions with the asymptotic behavior

v v 0 L 4 ( t , 2 t ) L 4 ( B ( 1 δ ) t ) t 1 2 +

as t, where v0(t,x)=2t and B(1δ)t:={x3:|x|<(1δ)t}.

Citation

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Roland Donninger. Anil Zenginoğlu. "Nondispersive decay for the cubic wave equation." Anal. PDE 7 (2) 461 - 495, 2014. https://doi.org/10.2140/apde.2014.7.461

Information

Received: 24 April 2013; Accepted: 22 August 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1295.35084
MathSciNet: MR3218816
Digital Object Identifier: 10.2140/apde.2014.7.461

Subjects:
Primary: 35L05, 35L71, 58J45
Secondary: 35Q75, 83C30

Rights: Copyright © 2014 Mathematical Sciences Publishers

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Vol.7 • No. 2 • 2014
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