Communications on Pure and Applied Analysis (CPAA)

On Fractional Schrödinger Equations in sobolev spaces

Pages: 2265 - 2282, Volume 14, Issue 6, November 2015      doi:10.3934/cpaa.2015.14.2265

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Younghun Hong - University of Texas at Austin, United States (email)
Yannick Sire - Université Aix-Marseille, I2M, France (email)

Abstract: Let $\sigma \in (0,1)$ with $\sigma \neq \frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: \begin{eqnarray} i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0, u(0)=u_0\in H^s, \end{eqnarray} 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 [23]. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

Keywords:  Fractional Schrödinger, Sobolev spaces, Local and global well-posedness, ill-posedness.
Mathematics Subject Classification:  35A01, 35B35, 35B45, 35B65.

Received: December 2014;      Revised: June 2015;      Available Online: September 2015.