Abstract
This note studies the fourth-order generalized Hartree equation iu + Delta 2u +/- |u|p-2(J gamma* |u|p)u = 0, p > 2. Indeed, for both attractive and repulsive sign, the scattering is obtained in the inter-critical regime, which is given by 0 < sc < 2, where the critical Sobolev 4+gamma exponent is given by the equality kappa 2(p-1) Ilu(kappa 4', kappa t)IlHsc = Ilu(kappa t)IlHsc . 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 > 5. Moreover, one assumes that p > 2 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 ei center dot Delta 2u +/-, when t-+ +/- oo. This means that the source term has no effect for large time.