Sharp threshold of blow-up and scattering for the fractional Hartree equation
type: Item
instance of: scholarly article
title: Sharp threshold of blow-up and scattering for the fractional Hartree equation
zbMATH Open document ID: 1393.35193
author: Qing Guo
author: Shihui Zhu
published in: Journal of Differential Equations
publication date: 2017-12-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1705.08615
review text: This article presents the fractional Hartree equation \[ iu_t-(-\Delta)^su+(\frac{1}{| x|^\gamma}\ast| u|^2)u=0, \] where \(s\in(0,1)\), \(\gamma\in(2s,\min\{N,4s\})\) and \(x\in\mathbb R^N\). Hence, the authors consider the supercritical case, in the \(L^2\) sense; moreover, the equation possesses scale invariance. The results obtained are for the case of radial data. Here, a sharp blow-up/scattering dichotomy result is formulated. The fractional Laplacian present here is based on the fractional Schrödinger equation [\textit{N. Laskin}, ``Fractional Schrödinger equation'', Phys. Rev. E (3) 66, No. 5, Article ID 056108, 7 p. (2002; \url{doi:10.1103/PhysRevE.66.056108})].
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35C08
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 6820638
zbMATH Keywords: fractional Schrödinger equation
zbMATH Keywords: \(L^2\)-supercritical
zbMATH Keywords: scattering
zbMATH Keywords: blow-up
reviewed by: Joseph Shomberg
MaRDI profile type: Publication
OpenAlex ID: W2619278050
arXiv ID: 1705.08615
cites work: Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates
cites work: Numerical methods for computing ground states and dynamics of nonlinear relativistic Hartree equation for boson stars
cites work: Blowup for fractional NLS
cites work: An Extension Problem Related to the Fractional Laplacian
cites work: Q4434430
cites work: A direct method of moving planes for the fractional Laplacian
cites work: On finite time blow-up for the mass-critical Hartree equations
cites work: On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity
cites work: On the orbital stability of fractional Schrödinger equations
cites work: Mean field dynamics of boson stars
cites work: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian
cites work: Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbb{R}\)
cites work: Blowup for nonlinear wave equations describing boson stars
cites work: Existence and stability of standing waves for nonlinear fractional Schrödinger equations
cites work: Scattering for the focusingL2-supercritical andḢ2-subcritical biharmonic NLS equations
cites work: Semiclassical states for weakly coupled fractional Schrödinger systems
cites work: Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations
cites work: Decay estimates for a class of wave equations
cites work: Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity
cites work: A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation
cites work: Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation
cites work: Nonlinear fractional Schrödinger equations in one dimension
cites work: Q4309712
cites work: On nonlinear Schrödinger equations. II: \(H^ S\)-solutions and unconditional well-posedness
cites work: Endpoint Strichartz estimates
cites work: Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrö\-dinger equation in the radial case
cites work: On the defect of compactness for the Strichartz estimates of the Schrödinger equations
cites work: Fractional quantum mechanics and Lévy path integrals
cites work: Well-posedness for semi-relativistic Hartree equations of critical type
cites work: Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs
cites work: Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
cites work: Global well-posedness and scattering for the defocusing \(H^{\frac 1 2}\)-subcritical Hartree equation in \(\mathbb R^d\)
cites work: Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case
cites work: Global well-posedness and scattering for the mass-critical Hartree equation with radial data
cites work: Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations.
cites work: Stability of standing waves for the nonlinear fractional Schrödinger equation
cites work: On the blow-up solutions for the nonlinear fractional Schrödinger equation
DOI: 10.1016/J.JDE.2017.11.001
description: scientific article