TY - JOUR

T1 - ON ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO CUBIC NONLINEAR KLEIN-GORDON SYSTEMS IN ONE SPACE DIMENSION

AU - Masaki, Satoshi

AU - Segata, Jun Ichi

AU - Uriya, Kota

N1 - Funding Information:
Received by the editors October 4, 2021. 2020 Mathematics Subject Classification. Primary 35L71; Secondary 35A22, 35B40. Key words and phrases. Nonlinear Klein-Gordon equation, asymptotic behavior of solutions, long-range scattering, normalization of systems, matrix representation. The first author was supported by JSPS KAKENHI Grant Numbers JP17K14219, JP17H02854, JP17H02851, and JP18KK0386. The second author was supported by JSPS KAK-ENHI Grant Numbers JP17H02851, JP19H05597, and JP20H00118. The third author was supported by JSPS KAKENHI Grant Number JP19K145.
Publisher Copyright:
© 2022 by the authors under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0).

PY - 2022

Y1 - 2022

N2 - In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order.

AB - In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order.

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U2 - 10.1090/btran/116

DO - 10.1090/btran/116

M3 - Article

AN - SCOPUS:85141954112

SN - 2330-0000

VL - 9

SP - 517

EP - 563

JO - Transactions of the American Mathematical Society Series B

JF - Transactions of the American Mathematical Society Series B

ER -