TY - JOUR

T1 - Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions

AU - Masaki, Satoshi

AU - Segata, Jun ichi

AU - Uriya, Kota

PY - 2020/7

Y1 - 2020/7

N2 - In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed.

AB - In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed.

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U2 - 10.1016/j.matpur.2020.03.009

DO - 10.1016/j.matpur.2020.03.009

M3 - Article

AN - SCOPUS:85082724977

VL - 139

SP - 177

EP - 203

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

ER -