Remarks on scattering for nonlinear Schrödinger equations

Kenji Nakanishi, Tohru Ozawa

Research output: Contribution to journalArticle

48 Citations (Scopus)

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/ (√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.

Original languageEnglish
Pages (from-to)45-68
Number of pages24
JournalNonlinear Differential Equations and Applications
Volume9
Issue number1
DOIs
Publication statusPublished - 2002 Dec 1
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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