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Description
We study the nonlinear Schrödinger equation (NLS)
∂tu+iΔu=iλ|u|p−1u
in R1+n, where n≥3, p>1, and λ∈C. We prove that (NLS) is locally well-posed in Hs if 1<s<min{4;n/2} and max{1;s/2}<p<1+4/(n−2s). To obtain a good lower bound for p, we use fractional order Besov spaces for the time variable. The use of such spaces together with time cut-off makes it difficult to derive positive powers of time length from nonlinear estimates, so that it is difficult to apply the contraction mapping principle. For the proof we improve Pecher's inequality (1997), which is a modification of the Strichartz estimate, and apply this inequality to the nonlinear problem together with paraproduct formula.
Subject
Nonlinear Schrödinger equations
well-posedness
Besov spaces
Journal Title
Communications on Pure & Applied Analysis
Volume
18
Issue
3
Start Page
1359
End Page
1374
ISSN
1534-0392
ISSN(Online)
1553-5258
Published Date
2019
DOI
Publisher
American Institute of Mathematical Sciences
NII Type
Journal Article
Format
PDF
Text Version
出版社版
OAI-PMH Set
Faculty of Science and Engineering
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