Small data scattering of semirelativistic Hartree equation

Abstract

In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension 3. The Hartree type nonlinearity is [V∗|u|2]u and the potential V which generalizes the Yukawa has some growth condition. We show that the solution scatters to linear solution if an initial data given in Hs,1 is sufficiently small and s>14. Here, Hs,1 is Sobolev type space taking in angular regularity with norm defined by ‖φ‖Hs,1=‖φ‖Hs+‖∇Sφ‖Hs. To establish the results we employ the recently developed Strichartz estimate which is Lθ2-averaged on the unit sphere S2 and construct the resolution space based on Up-Vp space.