Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 101, pp. 1-23.

Title: Global well-posedness for  Klein-Gordon-Hartree and fractional Hartree equations
       on modulation spaces
       
Author: Divyang G. Bhimani (Indian Inst. of Science Education and Research, Pune, India)

Abstract:
We study the Cauchy problems for the  Klein-Gordon (HNLKG), wave (HNLW),
and  Schrodinger (HNLS) equations  with cubic convolution (of Hartree type) nonlinearity.
Some global well-posedness and  scattering  are obtained for the (HNLKG) and (HNLS)
with small Cauchy data in some modulation spaces. Global well-posedness for fractional
Schrodinger (fNLSH) equation with Hartree type nonlinearity is obtained with
Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS)  and
(HNLKG) with rough data in modulation spaces  is shown.  As a consequence, we get local
and global well-posedness and scattering in  larger than usual $L^p$-Sobolev spaces.

Submitted April 30, 2021. Published December 21, 2021.
Math Subject Classifications: 35L71, 35Q55, 42B35, 35A01.
Key Words: Klein-Gordon-Hartree equation; fractional Hartree equation;
           wave-Hartree equation; well-posedness; modulation spaces; small initial data.