On the global Cauchy problem for the Hartree equation with rapidly decaying initial data

Abstract

This paper is concerned with the Cauchy problem for the Hartree equation on ${\mathbb{R}}^n,n\in \mathbb{N}$ with the nonlinearity of type $(|\cdot {|}^{-\gamma }⁎|u{|}^2)u,0<\gamma <n$. It is shown that a global solution with some twisted persistence property exists for data in the space $L^p\cap L^2,1\le p\le 2$ under some suitable conditions on γ and spatial dimension $n\in \mathbb{N}$. It is also shown that the global solution u has a smoothing effect in terms of spatial integrability in the sense that the map $t\mapsto u(t)$ is well defined and continuous from $\mathbb{R}\setminus \{0\}$ to $L^{p^{\prime }}$, which is well known for the solution to the corresponding linear Schrödinger equation. Local and global well-posedness results for hat $L^p$-spaces are also presented. The local and global results are proved by combining arguments by Carles–Mouzaoui with a new functional framework introduced by Zhou. Furthermore, it is also shown that the global results can be improved via generalized dispersive estimates in the case of one space dimension.