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

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

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