On the scattering problem of mass-subcritical Hartree equation

Abstract

In this article, we consider mass-subcritical Hartree equation. Scattering problem is treated in the framework of weighted spaces. We first establish basic properties such as local-wellposedness and criteria for finite-time blowup and scattering. Then, the first result is that uniform in time bound in critical weighted norm implies scattering. The proof is based on the concentration compactness/rigidity argument initiated by Kenig and Merle. By using the argument, existence of a threshold solution between small scattering solutions and other solutions is also deduced for the focusing model, which is the second result. The threshold is neither ground state nor any other standing wave solutions, as is known for the power type NLS equation.