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Lifespan of Solutions to Nonlinear Schrödinger Equations with General Homogeneous Nonlinearity of the Critical Order

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Advances in Harmonic Analysis and Partial Differential Equations

Part of the book series: Trends in Mathematics ((TM))

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Abstract

This paper is concerned with the upper bound of the lifespan of solutions to nonlinear Schrödinger equations with general homogeneous nonlinearity of the critical order. In Masaki and the first author (Differ Integr Equ 32(3–4):121–138, 2019), they obtain the upper bound of the lifespan of solutions to our equation via a test function method introduced by Zhang (Duke Math J 97(3):515–539, 1999; C R Acad Sci Paris Sér I Math 333(2):109–114, 2001). Their nonlinearity contains a non-oscillating term |u|1+2∕d which causes difficultly for constructing an even small data global solution. The non-oscillating term corresponds to the L 1-scaling critical. In this paper, it turns out that the upper bound can be refined by employing a unified test function by Ikeda and the second author (Nonlinear Anal 182:57–74, 2019).

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Acknowledgements

The first author was supported by JSPS KAKENHI grant number 19K14580. The second author was supported by JSPS KAKENHI grant number 18K13445.

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Authors and Affiliations

  1. Teacher Training Courses, Faculty of Education, Kagawa University, Takamatsu, Kagawa, Japan

    Hayato Miyazaki

  2. Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda-shi, Chiba, Japan

    Motohiro Sobajima

Authors
  1. Hayato Miyazaki
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  2. Motohiro Sobajima
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Corresponding author

Correspondence to Hayato Miyazaki .

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Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Vladimir Georgiev

  2. Department of Applied Physics, Waseda University, Tokyo, Japan

    Tohru Ozawa

  3. Department of Mathematics, Ghent University, Gent, Belgium

    Michael Ruzhansky

  4. Department of Mathematics, University of Stuttgart, Stuttgart, Germany

    Jens Wirth

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Miyazaki, H., Sobajima, M. (2020). Lifespan of Solutions to Nonlinear Schrödinger Equations with General Homogeneous Nonlinearity of the Critical Order. In: Georgiev, V., Ozawa, T., Ruzhansky, M., Wirth, J. (eds) Advances in Harmonic Analysis and Partial Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-58215-9_7

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