Advances in Differential Equations

On the focusing energy-critical fractional nonlinear Schrödinger equations

Yonggeun Cho, Gyeongha Hwang, and Tohru Ozawa

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Abstract

We consider the fractional nonlinear Schrödinger equation (FNLS) with non-local dispersion $|\nabla|^{\alpha}$ and focusing energy-critical Hartree type nonlinearity $[-(|x|^{-2{\alpha}}*|u|^2)u]$. We first establish a global well-posedness of radial case in energy space by adopting Kenig-Merle arguments [20] when the initial energy and initial kinetic energy are less than those of ground state, respectively. We revisit and highlight long time perturbation, profile decomposition and localized virial inequality. As an application of the localized virial inequality, we provide a proof for finite time blowup for energy critical Hartree equations via commutator technique introduced in [2].

Article information

Source
Adv. Differential Equations Volume 23, Number 3/4 (2018), 161-192.

Dates
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ade/1513652445

Subjects
Primary: M35Q55 35Q40: PDEs in connection with quantum mechanics

Citation

Cho, Yonggeun; Hwang, Gyeongha; Ozawa, Tohru. On the focusing energy-critical fractional nonlinear Schrödinger equations. Adv. Differential Equations 23 (2018), no. 3/4, 161--192. https://projecteuclid.org/euclid.ade/1513652445


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