Remarks on scattering for nonlinear Schrödinger equations

Kenji Nakanishi, Tohru Ozawa

研究成果: Article

44 引用 (Scopus)

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/ (√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.

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
9
発行部数1
DOI
出版物ステータスPublished - 2002
外部発表Yes

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Sobolev spaces
Nonlinear equations
Nonlinear Equations
Scattering
Maximal Regularity
Sums of Powers
Global Analysis
Weighted Spaces
Sobolev Spaces
Critical value
Completeness
Restriction
Distinct

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

これを引用

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AU - Nakanishi, Kenji

AU - Ozawa, Tohru

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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

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