The radial bi-harmonic generalized Hartree equation revisited

This note studies the fourth-order generalized Hartree equation$ i\dot u+\Delta^2 u\pm|u|^{p-2}(J_\gamma*|u|^p)u = 0,\quad p\geq2. $Indeed, for both attractive and repulsive sign, scattering is obtained in inter-critical regime, which given by $ 0<s_c<2 $, where critical Sobolev exponent equality \kappa^\frac{4+\gamma}{2(p-1)}\|u(\kappa^4\cdot,\kappa t)\|_{\dot H^{s_c}} \|u(\kappa $. In focusing follows method due to Dodson Murphy (Proc. Am. Math. Soc., 145, no. 11 (2017), 4859-4867). approach based on a criteria Morawetz estimate. avoids concentration-compactness requires heavy machinery order obtain desired space-time bounds. The Kenig-Merle road-map was used first author, previous paper, scattering. One assumes here that data spherically symmetric. condition will be removed paper progress. defocusing decay of solutions some Lebesgue norms coupled with prove estimates, one assume space dimension N\geq5 Moreover, p\geq2 avoid singularity source term. energy implies global considered equation are asymptotic e^{i\cdot\Delta^2}u_\pm when t\to\pm\infty means term has no effect large time.