### Abstract

We study the global Cauchy problem and scattering problem for the semirelativistic Hartree-type equation in ℝ^{n}, n ≥ 1, with nonlocal nonlinearity F(u) = λ(|x|^{-γ} * |u| ^{2})u, 0 < γ < n. We prove the existence and uniqueness of global solutions for 0 < γ < 2n/n+1, n ≥ 2 or γ > 2, n ≥ 3, and the nonexistence of asymptotically-free solutions for 0 < γ ≤ 1, n ≥ 3. We also specify asymptotic behavior of solutions as the mass tends to zero and infinity.

Original language | English |
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Pages (from-to) | 1060-1074 |

Number of pages | 15 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 38 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2006 Dec 1 |

Externally published | Yes |

### Keywords

- Global solution
- Nonexistence of asymptotically free solutions
- Scattering
- Semirelativistic Hartree-type equation

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics

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## Cite this

Cho, Y., & Ozawa, T. (2006). On the semirelativistic Hartree-type equation.

*SIAM Journal on Mathematical Analysis*,*38*(4), 1060-1074. https://doi.org/10.1137/060653688