The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions

Jason Murphy

doi:10.3934/dcds.2014.34.733

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2014, Volume 34, Issue 2: 733-748. Doi: 10.3934/dcds.2014.34.733

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The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions

Jason Murphy^1,

1.
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555

Received: December 31, 2012

Revised: January 31, 2013

Published: January 2014

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Abstract

We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in dimensions $d\geq 4.$ In the spirit of Kenig and Merle [10], we combine a concentration-compactness approach with the Lin--Strauss Morawetz inequality to prove that if a solution $u$ is bounded in $\dot{H}^{1/2}$ throughout its lifespan, then $u$ is global and scatters.

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Mathematics Subject Classification: Primary: 35Q55.

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