Scattering for a Radial Defocusing Inhomogeneous Choquard Equation

This work considers the long time behavior of the solutions u∈C(R,H1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u\in C(\mathbb{R},H^{1})$\end{document} to the attractive Hartree equation iu˙+Δu=(Iα∗|⋅|−γ|u|p)|x|−γ|u|p−2u.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ i\dot{u}+\Delta u=(I_{\alpha }*|\cdot |^{-\gamma }|u|^{p})|x|^{-\gamma }|u|^{p-2}u. $$\end{document} Indeed, using a classical Morawetz estimate, the scattering of radial global solutions is proved in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document}-super-critical and H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{1}$\end{document}-sub-critical regimes.