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

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 $i{u}_t+(-\Delta {)}^{\frac{\alpha }{2}}u=\pm |u{|}^{\frac{2\alpha }{d}}u$, $u(0,\cdot )\in L^2$ 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.

21 pages, 2 figures

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