The radial bi-harmonic generalized Hartree equation revisited

Abstract

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