The Cauchy problem for the Hartree type equation in modulation spaces

We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F(u) = (K * vertical bar u vertical bar(2))u under a specified condition on potential K with Cauchy data in modulation spaces M-p,M-q(R-d). We establish global well-posedness results in M-1,M-1(R-d) when K(x) = lambda vertical bar x vertical bar(-gamma)(lambda subset of R, 0 < gamma < min{2, d/2}); in M-p,M-d(R-d) (1 <= q <= min{p, p'} where p' is the Holder conjugate of p is an element of [1, 2]) when K is in Fourier algebra FL1 (R-d), and local well-posedness result in M-p,M-1 (R-d) (1 <= p <= infinity) when K is an element of M-1,M-infinity (R-d). (C) 2015 Elsevier Ltd. All rights reserved.