Small analytic solutions to the Hartree equation

摘要

We consider the Cauchy problem for the Hartree equation in space dimension d >= 3. We assume that the interaction potential V belongs to the weak L-d/2 space. We prove that if the initial data phi is sufficiently small in the L-2-sense and either phi or its Fourier transform F phi satisfies a real-analytic condition, then the solution u(t) is also real-analytic for any t not equal 0. We also prove that if phi and V satisfy some strong condition, then u(t) can be extended to an entire function on C-d for any t not equal 0. A part of our method is applicable to the final value problem. We remark that no L-2 smallness condition is imposed on first and higher order partial derivatives of phi and F phi. (C) 2015 Elsevier Inc. All rights reserved.