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
N1 - Funding Information:
We thank an anonymous referee for careful reading of the manuscript and for fruitful suggestions. S.M. was supported by JSPS KAKENHI Grant Numbers JP17K14219 , JP17H02854 , JP17H02851 , and JP18KK0386 . J.S. was supported by JSPS KAKENHI Grant Number JP17H02851 . K.U. was supported by JSPS KAKENHI Grant Number JP19K14578 .
Publisher Copyright:
© 2020 Elsevier Masson SAS
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.
UR - http://www.scopus.com/inward/record.url?scp=85082724977&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85082724977&partnerID=8YFLogxK
U2 - 10.1016/j.matpur.2020.03.009
DO - 10.1016/j.matpur.2020.03.009
M3 - Article
AN - SCOPUS:85082724977
SN - 0021-7824
VL - 139
SP - 177
EP - 203
JO - Journal des Mathematiques Pures et Appliquees
JF - Journal des Mathematiques Pures et Appliquees
ER -