Title: Sharp Threshold of Blow-up and Scattering for the fractional Hartree equation,
Abstract: We consider the fractional Hartree equation in the $L^2$-supercritical case,
and we find a sharp threshold of the scattering versus blow-up dichotomy for
radial data: If $
M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and
$M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}<M[Q]^{\frac{s-s_c}{s_c}}\|
Q\|^2_{\dot H^s}$, then the solution $u(t)$ is globally well-posed and
scatters; if $
M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and
$M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}>M[Q]^{\frac{s-s_c}{s_c}}\|
Q\|^2_{\dot H^s}$, the solution $u(t)$ blows up in finite time. This condition
is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global
but not scattering, which satisfies the equality in the above conditions. Here,
$Q$ is the ground-state solution for the fractional Hartree equation.