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  and smooth modification of phase correction by Ginibre and Ozawa .
All Science Journal Classification (ASJC) codes
- Applied Mathematics