Below and beyond the mass–energy threshold: scattering for the Hartree equation with radial data in $$\pmb {d \ge 5}$$

Abstract

We consider the Cauchy problem of the focusing $$\dot{H}^{1/2}$$-critical Hartree equation $$\begin{aligned} i u_{t} + \Delta u = - \Bigl ( |\cdot |^{-3} *|u(t)|^{2} \Bigr )u(t,x), \quad (t,x)\in {\mathbb {R}} \times {\mathbb {R}}^{d}. \end{aligned}$$By adapting the methods in Dodson and Murphy (Proc Am Math Soc 145(11):4859–4867, 2017), we shall prove a scattering result for solutions both below and beyond the mass–energy threshold M(Q)E(Q) and uniformly describe both cases the boundary of the scattering region by the ground state’s mass and potential energy product.