Global Cauchy problems for the Klein-Gordon, wave and fractional Schr\"odinger equations with Hartree nonlinearity on modulation spaces

Abstract

We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schr\"odinger (HNLS) equations with cubic convolution (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 Schr\"odinger (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. This improves known results in Sobolev spaces in some sense. As a consequence, we get local and global well-posedness and scattering in larger than usual $L^p-$Sobolev spaces and we could include wider class of Hartree type nonlinarity.