On Fractional Schrodinger Equations in sobolev spaces

Abstract

Let $\sigma \in (0,1)$ with $\sigma \ne \frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in ${\mathbb{R}}^d$: $$i{\partial }_{t}u+(-\Delta {)}^{\sigma }u+\mu |u{|}^{p-1}u=0,u(0)=u_0\in H^s,$$ where $(-\Delta {)}^{\sigma }$ is the Fourier multiplier of symbol $|\xi {|}^{2\sigma }$, and $\mu =\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

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