Critical nonlinear Schrödinger equations with data in homogeneous weighted L2 spaces

Chunhua Li; Nakao Hayashi

doi:10.1016/j.jmaa.2014.05.053

Critical nonlinear Schrödinger equations with data in homogeneous weighted L² spaces

Chunhua Li, Nakao Hayashi

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

We consider the initial value problem for nonlinear Schrödinger equations with the critical nonlinearities λ1|u|^2/nu, where Imλ₁≤0, when the space dimension n=1, 2. We prove the global existence of small solutions in homogeneous weighted L²(Rⁿ) spaces. It is shown that the small solutions decay uniformly like t^-n/2 for t>1 if Imλ₁=0. The higher uniform time decay rates t-n2(logt)-n2 for t>1 are obtained if Imλ₁<0.

Original language	English
Pages (from-to)	1214-1234
Number of pages	21
Journal	Journal of Mathematical Analysis and Applications
Volume	419
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2014.05.053
Publication status	Published - 2014 Nov 15
Externally published	Yes

Keywords

Critical nonlinearities
Nonlinear Schrödinger equations

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.jmaa.2014.05.053

Cite this

@article{c21449525a9145bcb8820932701f33cb,

title = "Critical nonlinear Schr{\"o}dinger equations with data in homogeneous weighted L2 spaces",

abstract = "We consider the initial value problem for nonlinear Schr{\"o}dinger equations with the critical nonlinearities λ1|u|2/nu, where Imλ1≤0, when the space dimension n=1, 2. We prove the global existence of small solutions in homogeneous weighted L2(Rn) spaces. It is shown that the small solutions decay uniformly like t-n/2 for t>1 if Imλ1=0. The higher uniform time decay rates t-n2(logt)-n2 for t>1 are obtained if Imλ1<0.",

keywords = "Critical nonlinearities, Nonlinear Schr{\"o}dinger equations",

author = "Chunhua Li and Nakao Hayashi",

note = "Funding Information: We are grateful to an unknown referee for many useful suggestions and comments. One of the authors (C.L.) would like to thank Professor T. Ogawa for his financial support on her stay in Japan. This work is completed during her stay in Japan in Feb. 2014. The work of N.H. is supported by JSPS KAKENHI Grant Numbers 24654034 , 25220702 .",

year = "2014",

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doi = "10.1016/j.jmaa.2014.05.053",

language = "English",

volume = "419",

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journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

publisher = "Academic Press Inc.",

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T1 - Critical nonlinear Schrödinger equations with data in homogeneous weighted L2 spaces

AU - Li, Chunhua

AU - Hayashi, Nakao

N1 - Funding Information: We are grateful to an unknown referee for many useful suggestions and comments. One of the authors (C.L.) would like to thank Professor T. Ogawa for his financial support on her stay in Japan. This work is completed during her stay in Japan in Feb. 2014. The work of N.H. is supported by JSPS KAKENHI Grant Numbers 24654034 , 25220702 .

PY - 2014/11/15

Y1 - 2014/11/15

N2 - We consider the initial value problem for nonlinear Schrödinger equations with the critical nonlinearities λ1|u|2/nu, where Imλ1≤0, when the space dimension n=1, 2. We prove the global existence of small solutions in homogeneous weighted L2(Rn) spaces. It is shown that the small solutions decay uniformly like t-n/2 for t>1 if Imλ1=0. The higher uniform time decay rates t-n2(logt)-n2 for t>1 are obtained if Imλ1<0.

AB - We consider the initial value problem for nonlinear Schrödinger equations with the critical nonlinearities λ1|u|2/nu, where Imλ1≤0, when the space dimension n=1, 2. We prove the global existence of small solutions in homogeneous weighted L2(Rn) spaces. It is shown that the small solutions decay uniformly like t-n/2 for t>1 if Imλ1=0. The higher uniform time decay rates t-n2(logt)-n2 for t>1 are obtained if Imλ1<0.

KW - Critical nonlinearities

KW - Nonlinear Schrödinger equations

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