Blow up versus scattering below the mass-energy threshold for the focusing NLH with potential

Abstract

In this paper, we study the blow up and scattering result of the solution to the focusing nonlinear Hartree equation with potential $$i\partial_t u +\Delta u - Vu = - (|\cdot|^{-3} \ast |u|^2)u, \qquad (t, x) \in \mathbb{R} \times \mathbb{R}^5 $$ in the energy space ${H}^1(\mathbb{R}^5)$ below the mass-energy threshold. The potential $V$ we considered is an extension of Kato potential in some sense. We extend the results of Meng [26] to nonlinear Hartree equation with potential $V$ under some conditions. By establishing a Virial-Morawetz estimate and a scattering criteria, we obtain the scattering theory based on the method from Dodson-Murphy [11].