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

2017/03/15 by Masaki, Satoshi, Segata, Jun-ichi
#81Q05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35L71 #Secondary 35B40

paper · doi:10.48550/arxiv.1703.04888

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.

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