### 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 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Nonlinear Analysis, Theory, Methods and Applications*,

*74*(6), 2094-2108. https://doi.org/10.1016/j.na.2010.11.015

**On Hartree equations with derivatives.** / Cho, Yonggeun; Lee, Sanghyuk; Ozawa, Tohru.

Research output: Contribution to journal › Article

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 74, no. 6, pp. 2094-2108. https://doi.org/10.1016/j.na.2010.11.015

}

TY - JOUR

T1 - On Hartree equations with derivatives

AU - Cho, Yonggeun

AU - Lee, Sanghyuk

AU - Ozawa, Tohru

PY - 2011/3/15

Y1 - 2011/3/15

N2 - We consider the Cauchy problem of two types of Hartree equations with exchangecorrelation correction terms: {iut - Δu = V k(u)u in ℝ1+n, k = 1,2,u(0) = φ in ℝn, n ≥ 1, where V1(u) = |x|-γ*1|u|2 + λ2|∇u| 2),V2(u) = |x|-γ *∥∇|δu|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.

AB - We consider the Cauchy problem of two types of Hartree equations with exchangecorrelation correction terms: {iut - Δu = V k(u)u in ℝ1+n, k = 1,2,u(0) = φ in ℝn, n ≥ 1, where V1(u) = |x|-γ*1|u|2 + λ2|∇u| 2),V2(u) = |x|-γ *∥∇|δu|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.

KW - Angular regularity

KW - Hartree equations with derivatives

KW - Smoothing effect

KW - Well-posedness

UR - http://www.scopus.com/inward/record.url?scp=79951813303&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79951813303&partnerID=8YFLogxK

U2 - 10.1016/j.na.2010.11.015

DO - 10.1016/j.na.2010.11.015

M3 - Article

VL - 74

SP - 2094

EP - 2108

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 6

ER -