Global well-posedness for the fourth-order Schrödinger equation with Hartree-type nonlinearity for Cauchy data in Lp

This paper is concerned with the Cauchy problem for the nonlinear fourth-order Schrödinger equation on R^{n}, with the nonlinearity of Hartree-type (| ·|^{-γ}∗|u|^{2} )u .It is shown that a global solution exists for initial data in the spaces L^{p} (p < 2) under somesuitable conditions on γ, n and p. The solution is established by using a data-decomposition argument, two kinds of generalized Strichartz estimates and a interpolation theorem.