Skip to main content
SpringerLink
Log in

Quenching rates for heat equations with coupled singular nonlinear boundary flux

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

We study finite time quenching for heat equations coupled via singular nonlinear boundary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent regions and appropriate initial data. This extends an original work by Pablo, Quirós and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.

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. Kawarada H. On solutions of initial boundary value problem for u t = u xx +1/(1-u). RIMS Kyoto Univ, 10: 729–736 (1975)

    Article  MathSciNet  Google Scholar 

  2. Chan C Y. New results in quenching. In: Proc 1st World Congress of Nonlinear Analysis. Berlin: Walter de Gruyter, 1996

    Google Scholar 

  3. Deng K, Xu M X. Quenching for a nonlinear diffusion equation with a singular boundary conditions. Z Angew Math Phys, 50: 574–584 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Deng K, Xu M X. On solutions of a singular diffusion equation. Nonlinear Anal, 41: 489–500 (2000)

    Article  MathSciNet  Google Scholar 

  5. Levine H A. Advances in quenching. In: Proc Internat Conf on Reaction-Diffusion Equations and Their Equilibrium States. Boston: Birkhäuser, 1992

    Google Scholar 

  6. Levine H A, Lieberman G M. Quenching of solutions of parabolic equations with nonlinear boundary conditions in several dimensions. J Reine Angew Math, 345: 23–38 (1983)

    MATH  MathSciNet  Google Scholar 

  7. Fila M, Levine H A. Quenching on the boundary. Nonlinear Anal, 21: 795–802 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. de Pablo A, Quirós F, Rossi J D. Nonsimultaneous quenching. Appl Math Lett, 15: 265–269 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ferreira R, de Pablo A, Quirós F, et al. Nonsimultaneous quenching in a system of heat equations coupled at the boundary. Z Angew Math Phys, 57: 1–9 (2006)

    Article  MathSciNet  Google Scholar 

  10. Chadam J M, Peirce A, Yin H M. The blowup property of solutions to some diffusion equations with localized nonlinear reactions. J Math Anal Appl, 169: 313–328 (1992)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China

    SiNing Zheng

  2. Department of Mathematics, China University of Mining and Technology (Beijing), Beijing, 100083, China

    XianFa Song

Authors
  1. SiNing Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. XianFa Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to SiNing Zheng.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10471013, 10771024)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zheng, S., Song, X. Quenching rates for heat equations with coupled singular nonlinear boundary flux. Sci. China Ser. A-Math. 51, 1631–1643 (2008). https://doi.org/10.1007/s11425-007-0178-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-007-0178-1

Keywords

MSC(2000)

Search

Navigation