Nondispersive decay for the cubic wave equation

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 $δ \in (0, 1)$ , we construct solutions with the asymptotic behavior

∥ v - v_{0} ∥_{L^{4} (t, 2 t) L^{4} (B_{(1 - δ) t})} ≲ t^{- \frac{1}{2} +}

as $t \to \infty$ , where $v_{0} (t, x) = \sqrt{2} ∕ t$ and $B_{(1 - δ) t} : = {x \in ℝ^{3} : | x | < (1 - δ) t}$ .

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

Vol. 7, No. 2, 2014

Roland Donninger and Anıl Zenginoğlu

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors