Mass concentration for the $L^2$-critical Nonlinear Schr\"odinger equations of higher orders

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.

Categories: math.AP