Sharp threshold of blow-up and scattering for the fractional Hartree equation

We consider the fractional Hartree equation in the L2-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If M[u0]s−scscE[u0]M[Q]s−scsc‖Q‖H˙s2, the solution u(t) blows up in finite time. This condition is sharp in the sense that the solitary wave solution eitQ(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.