Modified scattering for the klein-gordon equation with the critical nonlinearity in three dimensions

Satoshi Masaki, Jun Ichi Segata

Research output: Contribution to journalArticle

3 Citations (Scopus)

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: (□ + 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].

Original languageEnglish
Pages (from-to)1595-1611
Number of pages17
JournalCommunications on Pure and Applied Analysis
Volume17
Issue number4
DOIs
Publication statusPublished - 2018 Jul

Keywords

  • Asymptotic behavior
  • Final data problem
  • Modified scattering
  • Nonlinear Klein-Gordon equation
  • Scattering problem

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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