### 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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics
- Numerical Analysis

### Cite this

*SIAM Journal on Mathematical Analysis*,

*38*(4), 1060-1074. https://doi.org/10.1137/060653688

**On the semirelativistic Hartree-type equation.** / Cho, Yonggeun; Ozawa, Tohru.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 38, no. 4, pp. 1060-1074. https://doi.org/10.1137/060653688

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TY - JOUR

T1 - On the semirelativistic Hartree-type equation

AU - Cho, Yonggeun

AU - Ozawa, Tohru

PY - 2006

Y1 - 2006

N2 - We study the global Cauchy problem and scattering problem for the semirelativistic Hartree-type equation in ℝn, 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.

AB - We study the global Cauchy problem and scattering problem for the semirelativistic Hartree-type equation in ℝn, 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.

KW - Global solution

KW - Nonexistence of asymptotically free solutions

KW - Scattering

KW - Semirelativistic Hartree-type equation

UR - http://www.scopus.com/inward/record.url?scp=34249317023&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249317023&partnerID=8YFLogxK

U2 - 10.1137/060653688

DO - 10.1137/060653688

M3 - Article

VL - 38

SP - 1060

EP - 1074

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 4

ER -