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Energy Solutions to One-Dimensional Singular Parabolic Problems with \({ BV}\) Data are Viscosity Solutions

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Mathematics for Nonlinear Phenomena — Analysis and Computation (MNP2015 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 215))

Abstract

We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in BV, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga–Giga.

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Acknowledgements

Both authors thank the grant agencies for partial support during the preparation of this paper and the universities, which hosted them. AN was supported by a Grant-in-Aid for JSPS Fellows No. 25-7077 as well as by MNiSW through 2853/7.PR/2013/2 grant. PR was in part supported by the EC IRSES program “Flux”. Both authors were in part supported by NCN through grant 2011/01/B/ST1/01197. A part of the research was performed during AN visits to the University of Warsaw supported by the Warsaw Center of Mathematics and Computer Science through the program ‘Guests’. Some work was done during a PR visit to the University of Tokyo.

Both authors thank the referee for his/her constructive comments, which help to improve the text.

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Authors and Affiliations

  1. Graduate School of Mathematical Science, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo, 153-8914, Japan

    Atsushi Nakayasu

  2. Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 07-097, Warsaw, Poland

    Piotr Rybka

Authors
  1. Atsushi Nakayasu
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  2. Piotr Rybka
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Corresponding author

Correspondence to Atsushi Nakayasu .

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Editors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, Miyagi, Japan

    Yasunori Maekawa

  2. Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, Japan

    Shuichi Jimbo

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Nakayasu, A., Rybka, P. (2017). Energy Solutions to One-Dimensional Singular Parabolic Problems with \({ BV}\) Data are Viscosity Solutions. In: Maekawa, Y., Jimbo, S. (eds) Mathematics for Nonlinear Phenomena — Analysis and Computation. MNP2015 2015. Springer Proceedings in Mathematics & Statistics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-319-66764-5_9

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