On Fractional Schrödinger Equations in sobolev spaces

Younghun  Hong; Yannick  Sire

doi:10.3934/cpaa.2015.14.2265

Article Contents

2015, Volume 14, Issue 6: 2265-2282. Doi: 10.3934/cpaa.2015.14.2265

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On Fractional Schrödinger Equations in sobolev spaces

Younghun Hong^1, and
Yannick Sire^2,

1.
University of Texas at Austin
2.
Université Aix-Marseille, I2M

Received: November 30, 2014

Revised: May 31, 2015

Published: October 2015

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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.

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Mathematics Subject Classification: 35A01, 35B35, 35B45, 35B65.

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