Skip to main content
SpringerLink
Log in

Moving singularities for nonlinear diffusion equations in two space dimensions

  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

Abstract

We construct solutions with prescribed moving singularities for equations of porous medium type in two space dimensions. This complements a previous study of the problem where only dimensions higher than two were considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fila, M., Takahashi, J., Yanagida, E.: Solutions with moving singularities for equations of porous medium type. Nonlinear Anal. 179, 237–253 (2019)

    Article  MathSciNet  Google Scholar 

  2. Htoo, K.P.P., Takahashi, J., Yanagida, E.: Higher-dimensional moving singularities in a superlinear parabolic equation. J. Evol. Equ. 18, 1575–1593 (2018)

    Article  MathSciNet  Google Scholar 

  3. Htoo, K.P.P., Yanagida, E.: Singular solutions of a superlinear parabolic equation with homogeneous Neumann boundary conditions. Nonlinear Anal. 151, 96–108 (2017)

    Article  MathSciNet  Google Scholar 

  4. Kan, T., Takahashi, J.: On the profile of solutions with time-dependent singularities for the heat equation. Kodai Math. J. 37, 568–585 (2014)

    Article  MathSciNet  Google Scholar 

  5. Kan, T., Takahashi, J.: Time-dependent singularities in semilinear parabolic equations: behavior at the singularities. J. Differ. Equ. 260, 7278–7319 (2016)

    Article  MathSciNet  Google Scholar 

  6. Kan, T., Takahashi, J.: Time-dependent singularities in semilinear parabolic equations: Existence of solutions. J. Differ. Equ. 263, 6384–6426 (2017)

    Article  MathSciNet  Google Scholar 

  7. Karch, G., Schonbek, M.E., Schonbek, T.P.: Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete Contin. Dyn. Syst. 40, 189–206 (2020)

    Article  MathSciNet  Google Scholar 

  8. Karch, G., Zheng, X.: Time-dependent singularities in the Navier–Stokes system. Discrete Contin. Dyn. Syst. 35, 3039–3057 (2015)

    Article  MathSciNet  Google Scholar 

  9. Sato, S., Yanagida, E.: Solutions with moving singularities for a semilinear parabolic equation. J. Differ. Equ. 246, 724–748 (2009)

    Article  MathSciNet  Google Scholar 

  10. Sato, S., Yanagida, E.: Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete Contin. Dyn. Syst. 26, 313–331 (2010)

    Article  MathSciNet  Google Scholar 

  11. Sato, S., Yanagida, E.: Singular backward self-similar solutions of a semilinear parabolic equation. Discrete Contin. Dyn. Syst. Ser. S 4, 897–906 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Takahashi, J.: Existence of solutions with moving singularities for a semilinear heat equation with a critical exponent (preprint)

  13. Takahashi, J., Yamamoto, H.: Solutions with time-dependent singular sets for the heat equation with absorption. J. Differ. Equ. 266, 4061–4105 (2019)

    Article  MathSciNet  Google Scholar 

  14. Takahashi, J., Yanagida, E.: Time-dependent singularities in the heat equation. Commun. Pure Appl. Anal. 14, 969–979 (2015)

    Article  MathSciNet  Google Scholar 

  15. Takahashi, J., Yanagida, E.: Time-dependent singularities in a semilinear parabolic equation with absorption. Commun. Contemp. Math. 18, 1550077 (2016)

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Comenius University, 842 48, Bratislava, Slovakia

    Marek Fila & Petra Macková

  2. Department of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, 152-8552, Japan

    Jin Takahashi

  3. Department of Mathematics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan

    Eiji Yanagida

Authors
  1. Marek Fila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petra Macková
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jin Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eiji Yanagida
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marek Fila.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Dedicated to Michel Chipot on the occasion of his 70th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Marek Fila was supported in part by the Slovak Research and Development Agency under the Contract No. APVV-18-0308 and by the VEGA grant 1/0347/18. Jin Takahashi was supported by JSPS KAKENHI Early-Career Scientists (No. 19K14567). Eiji Yanagida was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (A) (No. 17H01095).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fila, M., Macková, P., Takahashi, J. et al. Moving singularities for nonlinear diffusion equations in two space dimensions. J Elliptic Parabol Equ 6, 155–169 (2020). https://doi.org/10.1007/s41808-020-00062-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41808-020-00062-0

Keywords

Mathematics Subject Classification

Search

Navigation