On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity

Abstract

Abstract. We study the Cauchy problem for the fractional Schrö dinger equation, where n b 1, m b 0, 1 < a < 2, and F stands for the nonlinearity of Hartree type F ðuÞ ¼ lðcðÁÞj Á j Àg à juj 2 Þu with l ¼G1, 0 < g < n,and 0 a c A L y ðR n Þ. We prove the existence and uniqueness of local and global solutions for certain a, g, l, c. We also remark on finite time blowup of solutions when l ¼ À1.

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

“…When γ = 0 and α ∈ (0, 1), the problem (1.1) is a nonlocal model known as nonlinear fractional Schrödinger equation which has also attracted much attentions recently [9,10,11,12,19,20,21,22,23,24,15,16]. The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics, which was derived by Laskin [29,30] as a result of extending the Feynman path integral, from the Brownian-like to Levy-like quantum mechanical paths.…”

On the unique continuation of solutions to non-local non-linear dispersive equations

“…The well-posedness of the IVP (1.19) has been considered in several publications, see for example [8], [23], [28], [30], [5] and references therein.…”

On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation

“…These weighted estimates are important to show the well-posedness below L 2 at least for the fractional Hartree equation (see [7]).…”

“…The local well-posedness for (1.1) in Sobolev spaces was studied in [21] (see also [7] for fractional Hartree equations). Note that the unitary group e −it(−∆) s enjoys several types of Strichartz estimates (see e.g.…”