Non-linear bi-harmonic Choquard equations

Abstract

This note studies the fourth-order Choquard equation \begin{document}$ i\dot u+\Delta^2 u\pm (I_\alpha *|u|^p)|u|^{p-2}u = 0 . $\end{document} In the mass super-critical and energy sub-critical regimes, a sharp threshold of global well-psedness and scattering versus finite time blow-up dichotomy is obtained.