Standing ring blowup solutions for cubic nonlinear Schrödinger equations

Zwiers, Ian

doi:10.2140/apde.2011.4.677

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Standing ring blowup solutions for cubic nonlinear Schrödinger equations

Ian Zwiers

Vol. 4 (2011), No. 5, 677–727

DOI: 10.2140/apde.2011.4.677

Abstract

For all dimensions $N \geq 3$ we prove there exist solutions to the focusing cubic nonlinear Schrödinger equations that blow up on a set of codimension two. The blowup set is identified both as the site of $L^{2}$ concentration and by a bounded supercritical norm outside any neighborhood of the set. In all cases, the global $H^{1}$ norm grows at the log-log rate.

Keywords

focusing, nonlinear Schrödinger equation, supercritical, collapse, blowup rate, blowup set, codimension, regularity, log-log rate

Mathematical Subject Classification 2000

Primary: 35Q55

Secondary: 35B40

Milestones

Received: 5 February 2010

Revised: 27 September 2010

Accepted: 14 November 2010

Published: 16 February 2012

Proposed: Frank Merle

Authors

Ian Zwiers
Department of Mathematics University of British Columbia Vancouver, V6T 1Z2 Canada

Vol. 4, No. 5, 2011