Scattering for the focusing ${\dot H}^{1/2}$-critical Hartree equation with radial data

TL;DR

This paper proves that for the focusing ${ m f rac{1}{2}}$-critical Hartree NLS with radial data in 5D, solutions below a certain energy threshold are global and scatter, extending understanding of critical dispersive equations.

Contribution

It establishes scattering and global existence for the focusing ${ m f rac{1}{2}}$-critical Hartree NLS with radial data under a specific energy condition, a new result in this critical regime.

Findings

Solutions below the energy threshold are global and scatter.

The threshold is related to the ground state solution Q.

The result applies to radial data in five dimensions.

Abstract

We investigate the focusing $\dot H^{1/2}$-critical nonlinear Schr\"{o}dinger equation (NLS) of Hartree type $i\partial_t u + \Delta u = -(|\cdot|^{-3} \ast |u|^2)u$ with $\dot H^{1/2}$ radial data in dimension $d = 5$. It is proved that if the maximal life-span solution obeys $\sup_{t}\big\||\nabla|^{{1/2}}u\big\|_2 < \frac{\sqrt{6}}{3} \big\||\nabla|^{{1/2}}Q\big\|_2$, where $Q$ is the positive radial solution to the elliptic equation(\ref{e14}). Then the solution is global and scatters.