Vol. 7, No. 2, 2014

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Nondispersive decay for the cubic wave equation

Roland Donninger and Anıl Zenginoğlu

Vol. 7 (2014), No. 2, 461–495
Abstract

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

(t2 + 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 v0L4(t,2t)L4(B(1δ)t) t1 2 +

as t , where v0(t,x) = 2t and B(1δ)t := {x 3 : |x| < (1 δ)t}.

Keywords
nonlinear wave equations, soliton resolution conjecture, hyperboloidal initial value problem, Kelvin coordinates
Mathematical Subject Classification 2010
Primary: 35L05, 58J45, 35L71
Secondary: 35Q75, 83C30
Milestones
Received: 24 April 2013
Accepted: 22 August 2013
Published: 30 May 2014
Authors
Roland Donninger
Department of Mathematics
École Polytechnique Fédérale de Lausanne
Station 8
CH-1015 Lausanne
Switzerland
Anıl Zenginoğlu
Theoretical Astrophysics
California Institute of Technology
M/C 350–17
Pasadena, CA 91125
United States