  Modulation spaces have received considerable interest recently as it is the
natural function spaces to consider low regularity Cauchy data for several
nonlinear evolution equations. We establish global well-posedness for 3D
Klein-Gordon-Hartree equation $$u_{tt}-\Delta u+u + ( |\cdot|^{-\gamma} \ast
|u|^2)u=0$$ with initial data in modulation spaces $M^{p, p'}_1 \times M^{p,p}$
for $p\in \left(2, \frac{54 }{27-2\gamma} \right),$ $2<\gamma<3.$ We implement
Bourgain's high-low frequency decomposition method to establish global
well-posedness, which was earlier used for classical Klein-Gordon equation.
This is the first result on low regularity for Klein-Gordon-Hartree equation
with large initial data in modulation spaces (which do not coincide with
Sobolev spaces).

Analysis of PDEs