Scattering for the radial 3D cubic wave equation

Dodson, Benjamin; Lawrie, Andrew

doi:10.2140/apde.2015.8.467

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Scattering for the radial 3D cubic wave equation

Benjamin Dodson and Andrew Lawrie

Vol. 8 (2015), No. 2, 467–497

DOI: 10.2140/apde.2015.8.467

Abstract

Consider the Cauchy problem for the radial cubic wave equation in $1 + 3$ dimensions with either the focusing or defocusing sign. This problem is critical in $Ḣ^{1 ∕ 2} \times Ḣ^{- 1 ∕ 2} (ℝ^{3})$ and subcritical with respect to the conserved energy. Here we prove that if the critical norm of a solution remains bounded on the maximal time interval of existence, then the solution must in fact be global in time and must scatter to free waves as $t \to \pm \infty$ .

Keywords

scattering, concentration compactness, double Duhamel

Mathematical Subject Classification 2010

Primary: 35L05

Milestones

Received: 20 May 2014

Revised: 12 October 2014

Accepted: 26 November 2014

Published: 10 May 2015

Authors

Benjamin Dodson
Department of Mathematics Johns Hopkins University 404 Krieger Hall 3400 N. Charles Street Baltimore, CA 21218 United States

Andrew Lawrie
Department of Mathematics University of California, Berkeley 970 Evans Hall #3840 Berkeley, CA 94720 United States

Vol. 8, No. 2, 2015