This paper is concerned with the Cauchy problem for the Hartree equation on with the nonlinearity of type . It is shown that a global solution with some twisted persistence property exists for data in the space under some suitable conditions on γ and spatial dimension . It is also shown that the global solution u has a smoothing effect in terms of spatial integrability in the sense that the map is well defined and continuous from to , which is well known for the solution to the corresponding linear Schrödinger equation. Local and global well-posedness results for hat -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.
Cite this article
Ryosuke Hyakuna, On the global Cauchy problem for the Hartree equation with rapidly decaying initial data. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 4, pp. 1081–1104DOI 10.1016/J.ANIHPC.2018.11.004