The energy-critical inhomogeneous generalized Hartree equation in 3D

TL;DR

This paper proves global well-posedness and scattering for the 3D energy-critical inhomogeneous generalized Hartree equation with non-radial data, introducing new techniques to handle potential and nonlocal nonlinearities.

Contribution

It establishes the first non-radial scattering results for this class of inhomogeneous Hartree equations using novel analytical methods.

Findings

Global well-posedness and scattering below the ground state for non-radial data.

Scattering results for the classical Hartree equation with radial data.

Scattering in the defocusing case with general data.

Abstract

The purpose of this work is to study the $3D$ energy-critical inhomogeneous generalized Hartree equation $$ i\pa_tu+\Delta u+|x|^{-b}(I_\alpha\ast|\cdot|^{-b}|u|^{p})|u|^{p-2}u=0,\;\ x\in\R^3, $$ where $p=3+\alpha-2b$. We establish global well-posedness and scattering below the ground state threshold with non-radial initial data in $\dot{H}^1$. To this end, we exploit the decay of the nonlinearity, which together with the Kenig-Merle roadmap, allows us to treat the non-radial case as the radial case. In this paper are introduced new techniques to overcome the challenges posed by the presence of the potential and the nonlocal nonlinear term of convolution type. In particular, we also show scattering for the classical generalized Hartree equation ($b=0$) assuming radial data. Additionally, in the defocusing case, we show scattering with general data. We believe that the ideas developed here are robust and can be applicable to other types of nonlinear Hartree equations. In the introduction, we discuss some open problems.