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

Satoshi Masaki; Jun-ichi Segata

doi:10.3934/cpaa.2018076

Article Contents

2018, Volume 17, Issue 4: 1595-1611. Doi: 10.3934/cpaa.2018076

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

Satoshi Masaki^1, , and
Jun-ichi Segata^2,

1.
Department systems innovation, Graduate school of Engineering Science, Osaka University, 1-3, Machikaneyamacho, Toyonaka, Osaka 560-8531, Japan
2.
Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan

* Corresponding author
* Corresponding author

Received: February 2017

Revised: July 2017

Published: April 2018

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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: $(\Box+1)u = λ|u|^{2/3}u$, $t∈\mathbb{R}$, $x∈\mathbb{R}^{3}$, where $\Box = \partial_{t}^{2}-Δ$ is d'Alembertian. We prove that for a given asymptotic profile $u_{\mathrm{ap}}$, there exists a solution $u$ to (NLKG) which converges to $u_{\mathrm{ap}}$ as $t\to∞$. Here the asymptotic profile $u_{\mathrm{ap}}$ 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].

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Mathematics Subject Classification: Primary: 35L71; Secondary: 35B40, 81Q05.

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References

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