Scattering for the focusing ${\dot H}^{1/2}$-critical Hartree equation with radial data
We investigate the focusing $\dot H^{1/2}$-critical nonlinear Schrödinger equation (NLS) of Hartree type $i\partial_t u + Δu = -(|\cdot|^{-3} \ast |u|^2)u$ 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.
math.AP↗