Scattering for the focusing [formula omitted]-critical Hartree equation in energy space

Abstract

We investigate the focusing nonlinear Schrödinger equation (NLS) of Hartree type i ∂ t u + Δ u = − ( | ⋅ | 2 − d ∗ | u | 2 ) u in R 5 with initial data in energy space H 1 . If M [ u 0 ] E [ u 0 ] < M [ Q ] E [ Q ] , ‖ u 0 ‖ 2 ‖ ∇ u 0 ‖ 2 < ‖ Q ‖ 2 ‖ ∇ Q ‖ 2 . Then the solution with initial data u 0 is global and scatters. Here Q is the ground state solution. Moreover, we show that if M [ u 0 ] E [ u 0 ] < M [ Q ] E [ Q ] , but ‖ u 0 ‖ 2 ‖ ∇ u 0 ‖ 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.