Skip to main content
SpringerLink
Log in

Blow-Up Profile of Solutions in Parabolic Equations with Nonlocal Dirichlet Conditions

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

This paper deals with parabolic equations with nonlocal Dirichlet boundary conditions. Critical exponent and simultaneous blow-up criteria are studied for blow-up solutions. Moreover, we determine uniform blow-up profiles and blow-up set to all kinds of simultaneous blow-up solutions. It is interesting that large weighted functions could provide sufficient help to the occurring of blow-up of solutions without conditions on initial data.

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. Ding, J.T., Kou, W.: Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions. J. Math. Anal. Appl. 470, 1–15 (2019)

    Article  MathSciNet  Google Scholar 

  2. Feng, M., Zhou, J.: Global existence and blow-up of solutions to a nonlocal parabolic equation with singular potential. J. Math. Anal. Appl. 464, 1213–1242 (2018)

    Article  MathSciNet  Google Scholar 

  3. Kong, L.H., Wang, M.X.: Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries. Sci. Chin. Ser. A. 50, 1251–1266 (2007)

    Article  MathSciNet  Google Scholar 

  4. Li, F.C., Huang, S.X., Xie, C.H.: Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discr. Contin. Dyn. Syst. 9, 1519–1532 (2003)

    Article  MathSciNet  Google Scholar 

  5. Lin, Z.G., Liu, Y.R.: Uniform blow-up profiles for diffusion equations with nonlocal source and nonlocal boundary. Acta Math. Sci. Ser. B. 24, 443–450 (2004)

    Article  MathSciNet  Google Scholar 

  6. Liu, B.C., Dong, M.Z., Li, F.J.: Asymptotic properties of blow-up solutions in reaction-diffusion equations with nonlocal boundary flux. Z. Angew. Math. Phys. 69, 22 (2018)

    Article  MathSciNet  Google Scholar 

  7. Souplet, P.: Uniform blow-up profiles and boundary behaviour for diffusion equations with nonlocal nonlinear source. J. Differ. Equ. 153, 374–406 (1999)

    Article  Google Scholar 

  8. Souplet, P.: Blow up in nonlocal reaction–diffusion equations. SIAM J. Math. Anal. 29, 1301–1334 (1998)

    Article  MathSciNet  Google Scholar 

  9. Zhou, J.: Blow-up and lifespan of solutions to a nonlocal parabolic equation at arbitrary initial energy level. Appl. Math. Lett. 78, 118–125 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This paper is supported by Shandong Provincial Natural Science Foundation, China.

Author information

Authors and Affiliations

  1. College of Science, China University of Petroleum, Qingdao, 266555, Shandong Province, People’s Republic of China

    Bingchen Liu, Changcheng Zhang & Yu Wei

Authors
  1. Bingchen Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changcheng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bingchen Liu.

Additional information

Communicated by Ali Abkar.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, B., Zhang, C. & Wei, Y. Blow-Up Profile of Solutions in Parabolic Equations with Nonlocal Dirichlet Conditions. Bull. Iran. Math. Soc. 46, 1437–1453 (2020). https://doi.org/10.1007/s41980-019-00334-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-019-00334-z

Keywords

Mathematics Subject Classification

Search

Navigation