Scattering for the focusing Ḣ1/2-critical Hartree equation in energy space

We investigate the focusing nonlinear Schrödinger equation (NLS) of Hartree type i∂tu+Δu=−(|⋅|2−d∗|u|2)u in R5R5 with initial data in energy space H1H1. If M[u0]E[u0]‖Q‖2‖∇Q‖2‖u0‖2‖∇u0‖2>‖Q‖2‖∇Q‖2, then the corresponding solution will blow up in finite time. The argument of this work is based on the linear profile decomposition, in the spirit of Kenig–Merle.