Scattering for the Mass-Critical Nonlinear Klein–Gordon Equations in Three and Higher Dimensions

Xing Cheng; Zihua Guo; Satoshi Masaki

doi:10.1007/s10013-023-00616-4

Scattering for the Mass-Critical Nonlinear Klein–Gordon Equations in Three and Higher Dimensions

Xing Cheng, Zihua Guo, Satoshi Masaki

School of Mathematics

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

Abstract

In this paper we consider the mass-critical nonlinear Klein–Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method developed by C. E. Kenig and F. Merle. The main novelty from the work of R. Killip, B. Stovall, and M. Visan (Trans. Amer. Math. Soc. 364, 2012) is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrödinger equation when the nonlinearity is not algebraic.

Original language	English
Pages (from-to)	869-909
Number of pages	41
Journal	Vietnam Journal of Mathematics
Volume	51
Issue number	4
DOIs	https://doi.org/10.1007/s10013-023-00616-4
Publication status	Published - Oct 2023

Keywords

Klein–Gordon equations
Large scale profile
Profile decomposition
Scattering
Well-posedness

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10.1007/s10013-023-00616-4Licence: CC BY

Cite this

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title = "Scattering for the Mass-Critical Nonlinear Klein–Gordon Equations in Three and Higher Dimensions",

abstract = "In this paper we consider the mass-critical nonlinear Klein–Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method developed by C. E. Kenig and F. Merle. The main novelty from the work of R. Killip, B. Stovall, and M. Visan (Trans. Amer. Math. Soc. 364, 2012) is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schr{\"o}dinger equation when the nonlinearity is not algebraic.",

keywords = "Klein–Gordon equations, Large scale profile, Profile decomposition, Scattering, Well-posedness",

author = "Xing Cheng and Zihua Guo and Satoshi Masaki",

note = "Funding Information: Xing Cheng has been supported by the NSF of Jiangsu Province (Grant No. BK20221497) and “the Fundamental Research Funds for the Central Universities{"} (No.B210202147). Zihua Guo was supported in part by the ARC project (No. DP200101065). Satoshi Masaki was supported by JSPS KAKENHI Grant Numbers JP17K14219, JP17H02854, JP17H02851, and JP18KK0386. The authors are grateful to Professor Nakanishi for helpful discussion. Xing Cheng wish to thank Professor Stefanov for the interest he has taken in this work and for fruitful discussions. Publisher Copyright: {\textcopyright} 2023, The Author(s).",

year = "2023",

month = oct,

doi = "10.1007/s10013-023-00616-4",

language = "English",

volume = "51",

pages = "869--909",

journal = "Vietnam Journal of Mathematics",

issn = "2305-221X",

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number = "4",

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TY - JOUR

T1 - Scattering for the Mass-Critical Nonlinear Klein–Gordon Equations in Three and Higher Dimensions

AU - Cheng, Xing

AU - Guo, Zihua

AU - Masaki, Satoshi

N1 - Funding Information: Xing Cheng has been supported by the NSF of Jiangsu Province (Grant No. BK20221497) and “the Fundamental Research Funds for the Central Universities" (No.B210202147). Zihua Guo was supported in part by the ARC project (No. DP200101065). Satoshi Masaki was supported by JSPS KAKENHI Grant Numbers JP17K14219, JP17H02854, JP17H02851, and JP18KK0386. The authors are grateful to Professor Nakanishi for helpful discussion. Xing Cheng wish to thank Professor Stefanov for the interest he has taken in this work and for fruitful discussions. Publisher Copyright: © 2023, The Author(s).

PY - 2023/10

Y1 - 2023/10

N2 - In this paper we consider the mass-critical nonlinear Klein–Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method developed by C. E. Kenig and F. Merle. The main novelty from the work of R. Killip, B. Stovall, and M. Visan (Trans. Amer. Math. Soc. 364, 2012) is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrödinger equation when the nonlinearity is not algebraic.

AB - In this paper we consider the mass-critical nonlinear Klein–Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method developed by C. E. Kenig and F. Merle. The main novelty from the work of R. Killip, B. Stovall, and M. Visan (Trans. Amer. Math. Soc. 364, 2012) is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrödinger equation when the nonlinearity is not algebraic.

KW - Klein–Gordon equations

KW - Large scale profile

KW - Profile decomposition

KW - Scattering

KW - Well-posedness

UR - https://www.scopus.com/pages/publications/85164468654

U2 - 10.1007/s10013-023-00616-4

DO - 10.1007/s10013-023-00616-4

M3 - Article

AN - SCOPUS:85164468654

SN - 2305-221X

VL - 51

SP - 869

EP - 909

JO - Vietnam Journal of Mathematics

JF - Vietnam Journal of Mathematics

IS - 4

ER -

Scattering for the Mass-Critical Nonlinear Klein–Gordon Equations in Three and Higher Dimensions

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Keywords

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Nonlinear harmonic analysis and dispersive partial differential equations

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Scattering for the Mass-Critical Nonlinear Klein–Gordon Equations in Three and Higher Dimensions

Abstract

Keywords

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Nonlinear harmonic analysis and dispersive partial differential equations

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