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 Rn,n∈N with the nonlinearity of type (|⋅|−γ⁎|u|2)u,0<γ<n. It is shown that a global solution with some twisted persistence property exists for data in the space Lp∩L2,1≤p≤2 under some suitable conditions on γ and spatial dimension n∈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↦u(t) is well defined and continuous from R∖{0} to Lp′, which is well known for the solution to the corresponding linear Schrödinger equation. Local and global well-posedness results for hat Lp-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.