Modulation spaces and non-linear Hartree type equations

Abstract

We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F(u)=(k⋆|u|2)u under a specified condition on potential k with Cauchy data in modulation spaces Mp,q(Rn). We establish global well-posedness results in Mp,p(Rn) with 1≤p<2nn+ν, when k(x)=λ|x|ν(λ∈R,0<ν<min{2,n2}); in Mp,q(Rn) with 1≤q≤p≤2, when k∈M∞,1(Rn).