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

Abstract

We investigate the focusing ${\dot{H}}^{1/2}$-critical nonlinear Schrödinger 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 ${\sup }_t\Vert |\nabla {|}^{1/2}u{\Vert }_2<\frac{\sqrt{6}}{3}\Vert |\nabla {|}^{1/2}Q{\Vert }_2$, where $Q$ is the positive radial solution to the elliptic equation(\ref{e14}). Then the solution is global and scatters.

46 pages

References in corpus (4)

• Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
• Global well-posedness and scattering for the mass-critical Hartree equation with radial data
• The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
• Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation