Scattering for 2d semi-relativistic Hartree equations with short range potential

Abstract

We study the long-time behavior of small solutions to the semi-relativistic Hartree equation in two dimensions. The nonlinear term is convolved with the singular potential $|x{|}^{-\gamma }$ for $1<\gamma <2$, which is referred to as a short-range interaction potential in the sense of scattering phenomenon. We establish the scattering results for small solutions in a weighted space. In other words, we prove that the nonlinear solutions exist globally and behave asymptotically like a linear solution whenever the initial data is sufficiently small. To achieve this, we obtain time decay estimates for the nonlinear term which are integrable in time. A key observation is that the loss of time decay arising in the course of weighted energy estimates can be recovered by the space resonance method with the help of the null structure.