Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions
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- by Satoshi Masaki, Hayato Miyazaki and Kota Uriya PDF
- Trans. Amer. Math. Soc. 371 (2019), 7925-7947 Request permission
Abstract:
In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [SIAM J. Math. Anal. 50 (2018), pp. 3251–3270], the first and second authors consider one- and two-dimensional cases and give a sufficient condition on the nonlinearity so that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converge to a given asymptotic profile in a faster rate than in the lower dimensional cases. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in the aforementioned article. Moreover, we present a candidate for the second asymptotic profile of the solution.References
Additional Information
- Satoshi Masaki
- Affiliation: Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan
- MR Author ID: 823235
- Email: masaki@sigmath.es.osaka-u.ac.jp
- Hayato Miyazaki
- Affiliation: Advanced Science, Department of Integrated Science and Technology, National Institute of Technology, Tsuyama College, Tsuyama, Okayama, 708-8509, Japan
- MR Author ID: 1061120
- Email: miyazaki@tsuyama.kosen-ac.jp
- Kota Uriya
- Affiliation: Department of Applied Mathematics, Faculty of Science, Okayama University of Science, Okayama, Okayama, 700-0005, Japan
- MR Author ID: 1076837
- Email: uriya@xmath.ous.ac.jp
- Received by editor(s): September 17, 2017
- Received by editor(s) in revised form: February 7, 2018, and April 22, 2018
- Published electronically: March 7, 2019
- Additional Notes: The first author was partially supported by Sumitomo Foundation, Basic Science Research Projects No. 161145, and by JSPS, Grant-in-Aid for Young Scientists (B) 17K14219.
The second author was partially supported by the Overseas Research Fellowship Program by the National Institute of Technology. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7925-7947
- MSC (2010): Primary 35Q55, 35P25
- DOI: https://doi.org/10.1090/tran/7636
- MathSciNet review: 3955539