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Transactions of the American Mathematical Society

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Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions


Authors: Satoshi Masaki and Jun-ichi Segata
Journal: Trans. Amer. Math. Soc. 370 (2018), 8155-8170
MSC (2010): Primary 35L71; Secondary 35B40, 81Q05
DOI: https://doi.org/10.1090/tran/7262
Published electronically: July 5, 2018
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Abstract: In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein-Gordon equation (NLKG) in two space dimensions: $ (\Box +1)u=\lambda \vert u\vert u$, $ t\in \mathbb{R}$, $ x\in \mathbb{R}^{2}$, where $ \Box =\partial _{t}^{2}-\Delta $ is d'Alembertian. For a given asymptotic profile $ u_{\mathrm {ap}}$, we construct a solution $ u$ to (NLKG) which converges to $ u_{\mathrm {ap}}$ as $ t\to \infty $. 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 Fourier series expansion of the nonlinearity.


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Additional Information

Satoshi Masaki
Affiliation: Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka Osaka, 560-8531, Japan
Email: masaki@sigmath.es.osaka-u.ac.jp

Jun-ichi Segata
Affiliation: Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan
Email: segata@m.tohoku.ac.jp

DOI: https://doi.org/10.1090/tran/7262
Keywords: Scattering problem
Received by editor(s): November 25, 2016
Received by editor(s) in revised form: April 17, 2017
Published electronically: July 5, 2018
Additional Notes: The first author was partially supported by the Sumitomo Foundation, Basic Science Research Projects No. 161145.
The second author was partially supported by JSPS, Grant-in-Aid for Young Scientists (A) 25707004.
Article copyright: © Copyright 2018 American Mathematical Society

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