The focusing NLS on exterior domains in three dimensions

Kai Yang

doi:10.3934/cpaa.2017112

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2017, Volume 16, Issue 6: 2269-2297. Doi: 10.3934/cpaa.2017112

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The focusing NLS on exterior domains in three dimensions

Kai Yang

Department of Mathematics, University of Iowa, Iowa city, IA 52242, USA

Received: April 30, 2016

Revised: October 31, 2016

Published: September 2017

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Abstract

We consider the Dirichlet problem of the focusing energy subcritical NLS outside a smooth compact strictly convex obstacle in dimension three. The critical space of our problem is $\dot{H}^s$ with $0<s<1$. In this paper, we proved that if the initial data $u_{0}$ satisfy $\Vert u_{0}\Vert _{2}^{1-s}\Vert \nabla u_{0}\Vert _{2}^{s}<\Vert \nabla Q\Vert _{2}^{s}\Vert Q\Vert _{2}^{1-s}$ and $ M(u_{0})^{1-s}E(u_{0})^{s}<M(Q)^{1-s}E(Q)^{s},$ then there exists a unique global solution which scatters in both time directions, where $Q$ denotes the ground state solution in the whole space case.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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