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

TL;DR

This paper investigates the conditions under which solutions to the focusing nonlinear Hartree equation with potential either blow up or scatter, extending previous results to include a broader class of potentials in five-dimensional space.

Contribution

It extends existing blow-up and scattering results to the nonlinear Hartree equation with a generalized potential in five dimensions, using Virial-Morawetz estimates and scattering criteria.

Findings

Established scattering criteria for the equation with potential

Extended results to a broader class of potentials including Kato-type

Provided conditions under which solutions blow up or scatter

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].