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

We consider the fractional Hartree equation in the L-2-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If M[u(0)](s-sc/sc) E[u(0)] < M[Q](s-sc/sc) E[Q] and M[u(0)](s-sc/sc) parallel to u(0)parallel to(2)(<(H)over dot>s) < M[Q](s-sc/sc) parallel to Q parallel to(2)(<(H)over dot>s), then the solution u(t) is globally well-posed and scatters; if M[u(0)](s-sc/sc) E[u(0)] < M[Q](s-sc/sc) E[Q] and M[u(0)](s-sc/sc) parallel to u(0)parallel to(2)(<(H)over dot>s) > M[Q](s-sc/sc)parallel to Q parallel to(2)(<(H)over dot>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, Qis the ground-state solution for the fractional Hartree equation. (C) 2017 Elsevier Inc. All rights reserved.