## Abstract

We study the scattering problem and asymptotics for large time of solutions to the Cauchy problem for the nonlinear schrödinger and Hartree type equations wit h subcritical nonlinearities [Formula] where the nonlinear interaction term is F(|u|^{2}) = V * |u|^{2}, V(x) = λ|x|^{− δ}, λ ∈ R, 0 < δ < 1 in the Hartree type case, or F(|u|^{2}) = λ|t|^{1 −δ}|u|^{2} in the case of the cubic nonlinear Schrödinger equation. We suppose that the initial data e^{β|x|}u_{0} ∈ L^{2}, β > 0 with sufficiently small norm ε = ||e^{β|x|}u_{0}||L^{2}. Then we prove the sharp decay estimate ||u(t)||L^{p} ≤ C ϵt 1/p − 1/2, for all t ≥ 1 and for every 2 ≤ p ≤ ∞. Furthermore we show that for 1/2 < δ < 1 there exists a unique final state û + ∈ L^{2} such that for all t ≥ 1 [Formula] and uniformly with respect to x [formula] where ϕ ^ denotes the Fourier transform of ϕ. Our results show that the regularity condition on the initial data which was assumed in the previous paper [9] is not needed. Also a smoothing effect for the solutions in an analytic function space is discussed.

Original language | English |
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Pages (from-to) | 651-667 |

Number of pages | 17 |

Journal | Hokkaido Mathematical Journal |

Volume | 27 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1998 |

Externally published | Yes |

## Keywords

- Nonlinear Schrödinger
- Scattering
- Subcritical case

## ASJC Scopus subject areas

- Mathematics(all)