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

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

2009-04-20

arxiv.org/abs/0904.3021v1

Mathematics

Analysis of PDEs

21 pages, 2 figures

Scientific paper

We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schr\"odinger 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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