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

TL;DR

This paper establishes the existence of solutions to the three-dimensional nonlinear Klein-Gordon equation with critical nonlinearity that asymptotically match a prescribed profile, using modified scattering techniques involving Fourier series and phase corrections.

Contribution

It introduces a novel approach combining Fourier series expansion and phase correction modifications to handle critical nonlinearity in the Klein-Gordon equation.

Findings

Existence of solutions with prescribed asymptotic behavior

Construction of approximate solutions using Fourier series and phase correction

Extension of modified scattering theory to critical nonlinearity case

Abstract

In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions. We prove that for a given asymptotic profile, there exists a solution to (NLKG) which converges to given asymptotic profile as t to infinity. Here the asymptotic profile 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-Ozawa.