### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 45-68 |

Number of pages | 24 |

Journal | Nonlinear Differential Equations and Applications |

Volume | 9 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

- Global existence
- Lorentz spaces
- Nonlinear Schrödinger equation
- Scattering
- Strichartz estimate

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

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

Research output: Contribution to journal › Article

*Nonlinear Differential Equations and Applications*, vol. 9, no. 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 -