Scattering Criteria for the Three-Dimensional Focusing Energy-Critical Generalized Hartree Equation

Abstract

Two nonradial scattering criteria are established for the three-dimensional focusing energy-critical generalized Hartree equation below the ground-state threshold, under a bounded-scale concentration--compactness reduction. The first criterion is formulated in terms of fixed-center occupation windows and quantitative bounds on the motion of the concentration center. The second is based on uniform low-frequency \(L^2\)-decay, which yields finite mass, precompactness in the inhomogeneous energy space, a zero-momentum normalization, and sublinear center drift. The proofs combine a localized Hartree virial identity with a critical Hardy--Littlewood--Sobolev estimate controlling the nonlocal tail.