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

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  • 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.
    Mathematics Subject Classification: Primary: 35Q55.


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