### Abstract

We consider the Cauchy problem of two types of Hartree equations with exchangecorrelation correction terms: {iu_{t} - Δu = V _{k}(u)u in ℝ^{1+n}, k = 1,2,u(0) = φ in ℝ^{n}, n ≥ 1, where V_{1}(u) = |x|^{-γ}*_{1}|u|^{2} + λ_{2}|∇u| ^{2}),V_{2}(u) = |x|^{-γ} *∥∇|^{δ}u|^{2}). We establish the well-posedness of Cauchy problems and show the smoothing effect of solutions for each 0 < γ < n and 0 ≤ δ ≤ 1.

Original language | English |
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Pages (from-to) | 2094-2108 |

Number of pages | 15 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 74 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2011 Mar 15 |

### Keywords

- Angular regularity
- Hartree equations with derivatives
- Smoothing effect
- Well-posedness

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

Cho, Y., Lee, S., & Ozawa, T. (2011). On Hartree equations with derivatives.

*Nonlinear Analysis, Theory, Methods and Applications*,*74*(6), 2094-2108. https://doi.org/10.1016/j.na.2010.11.015