SUT J. Math. 34 (1), 13-24, (January 1998) DOI: 10.55937/sut/991983736
Nakao Hayashi, Pavel I. Naumkin
KEYWORDS: Hartree type equations, long range potential, Asymptotics in time, 35Q55
We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations
where the nonlinear interaction term is , . We suppose that the initial data and the value is sufficiently small, where is an integer satisfying , and denotes the largest integer less than . Then we prove that there exists a unique final state such that for all
uniformly with respect to with the following decay estimate , for all and for every . Furthermore we show that for there exists a unique final state such that for all
and uniformly with respect to
where denotes the Fourier transform of the function . In  we assumed that , and showed the same results as in this paper. Here we show that we do not need regularity conditions on the initial data by showing the local existence theorem in lower order Sobolev spaces.