On the existence of global solutions of the Hartree equation for initial data in the modulation space [formula omitted

Abstract

In this paper, we study the Cauchy problem for Hartree type equationiut+uxx=[K⁎|u|2]u, with Cauchy data in modulation spaces Mp,q(R). We establish global well-posedness results in Mp,p′(R) when K(x)=λ|x|γ,(λ∈R,0<γ<1) with no smallness condition on initial data, where p′ is the Hölder conjugate of p. Our proof uses a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Grünrock and Chaichenets et al. to the modulation space setting and exploits polynomial growth of the Schrödinger propagator on modulation spaces.