TY - JOUR
T1 - Modified scattering for the klein-gordon equation with the critical nonlinearity in three dimensions
AU - Masaki, Satoshi
AU - Segata, Jun Ichi
N1 - Funding Information:
Acknowledgments. The authors are grateful to anonymous referees for careful reading the manuscript and for valuable comments. S.M. is partially supported by the Sumitomo Foundation, Basic Science Research Projects No. 161145. J.S. is partially supported by JSPS, Grant-in-Aid for Young Scientists (A) 25707004.
Publisher Copyright:
© 2018 American Institute of Mathematical Sciences. All rights reserved.
PY - 2018/7
Y1 - 2018/7
N2 - In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions: (□ + 1)u = λ|u|2/3u, t ∈ ℝ, x ∈ ℝ3, where □ = ∂2t - Δ is d'Alembertian. We prove that for a given asymptotic profile uap, there exists a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper [23] and smooth modification of phase correction by Ginibre and Ozawa [6].
AB - In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions: (□ + 1)u = λ|u|2/3u, t ∈ ℝ, x ∈ ℝ3, where □ = ∂2t - Δ is d'Alembertian. We prove that for a given asymptotic profile uap, there exists a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper [23] and smooth modification of phase correction by Ginibre and Ozawa [6].
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U2 - 10.3934/cpaa.2018076
DO - 10.3934/cpaa.2018076
M3 - Article
AN - SCOPUS:85045328854
SN - 1534-0392
VL - 17
SP - 1595
EP - 1611
JO - Communications on Pure and Applied Analysis
JF - Communications on Pure and Applied Analysis
IS - 4
ER -