On the existence of global solutions of the Hartree equation for initial data in the modulation space Mp(qR)

In this paper, we study the Cauchy problem for Hartree type equation iu(t) + u(xx) = [K * vertical bar u vertical bar(2)] with Cauchy data in modulation spaces Mp,q(R). We establish global well-posedness results in Mp,p' (R) when K(x) = lambda/vertical bar x vertical bar(gamma), (lambda is an element of R, 0 < gamma < 1) with no smallness condition on initial data, where p' is the Holder conjugate of p. Our proof uses a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Grunrock and Chaichenets et al. to the modulation space setting and exploits polynomial growth of the Schrodinger propagator on modulation spaces. (c) 2022 Elsevier Inc. All rights reserved.