### 抄録

We unify two distinct methods of the global analysis for the nonlinear Schrödinger equations, namely those in the Sobolev spaces and in the weighted spaces. Thus we can deal with various sums of power nonlinearies |u|^{p-1} u for 1 + 2/n < p < ∞, since the former works for p ≥ 1 + 4/N, while the latter for 1 + 2/n < p < 1 + 4/n. Even for a single power, our result is much simpler and slightly better than the previous ones as to restriction on the initial data. Moreover, we extend the result to the maximal regularity, thereby obtaining scattering at the lower critical value p = 1 + 8/ (√n^{2} + 4n + 36 + n + 2) for n ≥ 4. We also show the asymptotic completeness in FH^{1} without smallness for p ≥ l+8/( √ n^{2} + 12n + 4+n-2) and any n ∈ ℕ.

元の言語 | English |
---|---|

ページ（範囲） | 45-68 |

ページ数 | 24 |

ジャーナル | Nonlinear Differential Equations and Applications |

巻 | 9 |

発行部数 | 1 |

DOI | |

出版物ステータス | Published - 2002 |

外部発表 | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### これを引用

**Remarks on scattering for nonlinear Schrödinger equations.** / Nakanishi, Kenji; Ozawa, Tohru.

研究成果: Article

*Nonlinear Differential Equations and Applications*, 巻. 9, 番号 1, pp. 45-68. https://doi.org/10.1007/s00030-002-8118-9

}

TY - JOUR

T1 - Remarks on scattering for nonlinear Schrödinger equations

AU - Nakanishi, Kenji

AU - Ozawa, Tohru

PY - 2002

Y1 - 2002

N2 - We unify two distinct methods of the global analysis for the nonlinear Schrödinger equations, namely those in the Sobolev spaces and in the weighted spaces. Thus we can deal with various sums of power nonlinearies |u|p-1 u for 1 + 2/n < p < ∞, since the former works for p ≥ 1 + 4/N, while the latter for 1 + 2/n < p < 1 + 4/n. Even for a single power, our result is much simpler and slightly better than the previous ones as to restriction on the initial data. Moreover, we extend the result to the maximal regularity, thereby obtaining scattering at the lower critical value p = 1 + 8/ (√n2 + 4n + 36 + n + 2) for n ≥ 4. We also show the asymptotic completeness in FH1 without smallness for p ≥ l+8/( √ n2 + 12n + 4+n-2) and any n ∈ ℕ.

AB - We unify two distinct methods of the global analysis for the nonlinear Schrödinger equations, namely those in the Sobolev spaces and in the weighted spaces. Thus we can deal with various sums of power nonlinearies |u|p-1 u for 1 + 2/n < p < ∞, since the former works for p ≥ 1 + 4/N, while the latter for 1 + 2/n < p < 1 + 4/n. Even for a single power, our result is much simpler and slightly better than the previous ones as to restriction on the initial data. Moreover, we extend the result to the maximal regularity, thereby obtaining scattering at the lower critical value p = 1 + 8/ (√n2 + 4n + 36 + n + 2) for n ≥ 4. We also show the asymptotic completeness in FH1 without smallness for p ≥ l+8/( √ n2 + 12n + 4+n-2) and any n ∈ ℕ.

KW - Global existence

KW - Lorentz spaces

KW - Nonlinear Schrödinger equation

KW - Scattering

KW - Strichartz estimate

UR - http://www.scopus.com/inward/record.url?scp=0346386030&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346386030&partnerID=8YFLogxK

U2 - 10.1007/s00030-002-8118-9

DO - 10.1007/s00030-002-8118-9

M3 - Article

AN - SCOPUS:0346386030

VL - 9

SP - 45

EP - 68

JO - Nonlinear Differential Equations and Applications

JF - Nonlinear Differential Equations and Applications

SN - 1021-9722

IS - 1

ER -