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 $[V\ast |u{|}^2]u$ 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 $H^{s,1}$ is sufficiently small and $s>\frac{1}{4}$. Here, $H^{s,1}$ is Sobolev type space taking in angular regularity with norm defined by $\Vert \phi {\Vert }_{H^{s,1}}=\Vert \phi {\Vert }_{H^s}+\Vert {\nabla }_{\mathbb{S}}\phi {\Vert }_{H^s}$. To establish the results we employ the recently developed Strichartz estimate which is $L_{\theta }^2$-averaged on the unit sphere ${\mathbb{S}}^2$ and construct the resolution space based on $U^p$-$V^p$ space.

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