Mass concentration for the $L^2$-critical nonlinear Schrödingerequations of higher orders

Myeongju Chae; Sunggeum Hong; Sanghyuk Lee

doi:10.3934/dcds.2011.29.909

Article Contents

2011, Volume 29, Issue 3: 909-928. Doi: 10.3934/dcds.2011.29.909

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Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders

1.
Department of Applied Mathematics, Hankyong National University, Ansong 456-749
2.
Department of Mathematics, Chosun University, Gwangju 501-759
3.
School of Mathematical Sciences, Seoul National University, Seoul 151-742

Received: August 2009

Revised: September 2010

Published: September 2011

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Abstract

We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schrödinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-\Delta)^{\frac\alpha 2} u =\pm |u|^\frac{2\alpha}{d}u$, $u(0,\cdot)\in L^2$ for $\alpha >2$, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as $\alpha$ increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.

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Mathematics Subject Classification: Primary: 35B05, 35B30, 35B33; Secondary: 35Q55, 42B10.

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