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In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions: \begin{document}$(\Box+1)u = λ|u|^{2/3}u$\end{document} , \begin{document}$t∈\mathbb{R}$\end{document} , \begin{document}$x∈\mathbb{R}^{3}$\end{document} , where \begin{document}$\Box = \partial_{t}^{2}-Δ$\end{document} is d'Alembertian. We prove that for a given asymptotic profile \begin{document}$u_{\mathrm{ap}}$\end{document} , there exists a solution \begin{document}$u$\end{document} to (NLKG) which converges to \begin{document}$u_{\mathrm{ap}}$\end{document} as \begin{document}$t\to∞$\end{document} . Here the asymptotic profile \begin{document}$u_{\mathrm{ap}}$\end{document} 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 ].

Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions