## Abstract

We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations { iu _{t} = - 1/2Δu + f(|u| ^{2}u, (t, x) ∈ R × R ^{n}, u(0,x) = u _{0}(x), x ∈ R ^{n}, n ≥ 2, where the nonlinear interaction term is f(|u| ^{2}) = V * |u| ^{2}, V(x) = λ|x| ^{- δ} λ ∈ R,0 < δ <1. We suppose that the initial data u _{0} ∈ H ^{0.l} and the value ∈= ||u _{0}||H ^{0.1} is sufficiently small, where l is an integer satisfying l ≥ [n/2] + 3, and [s] denotes the largest integer less than s. Then we prove that there exists a unique final state u _{+} ∈ H ^{0,l-2} such that for all t > 1 u(t, x) = 1 û/(it) ^{n/2} + (x/t) exp(ix ^{2}/2t-it ^{1- δ}/1- δ f(|û+| ^{2})(x/t) + 0(1+t ^{1-2δ}))+0( ^{-n/2-δ}) uniformly with respect to x ∈ R ^{n} with the following decay estimate ||u(t) ||L ^{p} ≤ C∈t ^{n/p-n/2,} for all t ≥ 1 and for every 2 ≤ p ≤ ∞. Furthermore we show that for 1/2 < δ <1 there exists a unique final state u + ∈ H ^{0,1-2} such that for all t ≥ 1, ||u(t)-exp(-it ^{1-δ}/1- δ f(|û+| ^{2}(x/t)) U(t)u+||L ^{2} = 0(t ^{1-2 δ}), and uniformly with respect to x ∈ R ^{n} u(t, x) = 1/(it) ^{n/2} û+(x/t)exp(ix ^{2}/2t-it ^{1-} ^{δ}/1- δ f(|û +

Original language | English |
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Pages (from-to) | 13-24 |

Number of pages | 12 |

Journal | SUT Journal of Mathematics |

Volume | 34 |

Issue number | 1 |

Publication status | Published - 1998 Dec 1 |

Externally published | Yes |

## Keywords

- Asymptotics in time
- Hartree type equations
- Long range potential

## ASJC Scopus subject areas

- Mathematics(all)