2023/12/21 by Yang, Changhun
#35B40 #35P25 #35Q40 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.13606
We study the long time behavior of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is convolved with the singular potential |x|-γ for 1<γ<2, which is referred to as 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 should obtain time decay estimates for the nonlinear term which is integrable. A key observation is that the loss in time in the course of weighted energy estimates can be recovered by the space resonance method with the help of null structure.