The Cauchy Problem for Non-linear Higher Order Hartree Type Equation in Modulation Spaces

We study theCauchy problem forHartree equationwith cubic convolution nonlinearity F( u) = ( K | u| 2k) u under a specified condition on potential K with Cauchy data in modulation spacesMp, q ( Rn). We establish global well- posedness results inM1,1( Rn), when K( x) =. | x|. (.. R, 0 <. < min{2, n 2}), for k < n+ 2-. n; and local wellposedness results in M1,1( Rn), when K( x) =. | x|. (.. R, 0 <. < n), for k. N; in Mp, q ( Rn) with 1 = p = 4, 1 = q = 22k- 2 22k- 2- 1, k. N, when K. M 8, 1( Rn). Moreover, we also consider theCauchy problem for the non- linear higher order Hartree equations on modulation spaces Mp, 1( Rn), when K. M1,8 ( Rn) and show the existence of a unique global solution by using integrability of time decay factors of Strichartz estimates. As a consequence, we are able to deal with wider classes of a nonlinearity and a solution space.