Skip to main content
SpringerLink
Log in

Dead-core rates for the fast diffusion equation with a spatially dependent strong absorption

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

This paper deals with the dead-core rates problem for the fast diffusion equation with a spatially dependent strong absorption

$$u_t=(u^{m})_{xx}-x^{q}u^p, \quad(x,t)\in(0,1)\times(0,\infty),$$

where 0 < p < m < 1 and −1 < q < 0. By using the self-similar transformation technique and the Zelenyak method, we proved that the temporal dead-core rate is non-self-similar.

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. Andreucci D., Tedeev A.F.: Universal bounds at the blow-up time for nonlinear parabolic equations, Adv. Diff. Equa. 10, 89–120 (2005)

    MATH  MathSciNet  Google Scholar 

  2. Bandle C., Piro S.V.: Estimates for solutions of quasilinear problems with dead cores. Z. Angew. Math. Phys. 54, 815–821 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Bandle and I. Stakgold, The formation of the dead core in parabolic reaction-diffusion problems, Trans. Amer. Math. Soc. 286 (1984), 275–293.

  4. X. F. Chen, J. S. Guo, and B. Hu, Dead-core rates for the porous medium equation with a strong absorption, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), 1761–1774.

  5. A. Friedman and J. B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425–447.

  6. S. C. Fu, J. S. Guo, and C. C. Wu, Dead-core at time infinity for a heat equation with strong absorption, Taiwanese J. Math. 13 (2009), 1213–1227.

  7. V. A. Galaktionov and J. L. Vazquez, Extinction for a quasilinear heat equation with absorption I: Technique of Intersection Comparison, Comm. Partial Differential Equations 19 (1994), 1075–1106.

  8. V. A. Galaktionov and J. L. Vazquez, Extinction for a quasilinear heat equation with absorption II: A dynamical systems approach, Comm. Partial Differential Equations 19 (1994), 1107–1137.

  9. J. S. Guo and B. Hu, Quenching profile for a quasilinear parabolic equation, Quarterly Appl. Math. 58 (2000), 613–626.

  10. J. S. Guo, C. T. Ling, and P. Souplet, Non-self-similar dead-core rate for the fast diffusion equation with strong absorption, Nonlinearity 23 (2010), 657–673.

  11. J. S. Guo, H. Matano, and C. C. Wu, An application of braid group theory to the finite time dead-core rate, J. Evol. Equ. 10 (2010), 835–855.

  12. J. S. Guo and Ph. Souplet, Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up, Math. Ann. 331 (2005), 651–667.

  13. J. S. Guo and C. C. Wu, Finite time dead-core rate for the heat equation with a strong absorption, Tohoku Math. J. 60 (2008), 37–70.

  14. M. A. Herrero and J. L. L. Velázquez, Approaching an extinction point in one-dimension semilinear heat equations with strong absorption, J. Math. Anal. Appl. 170 (1992), 353–381.

  15. Qi Y.W.: The critical exponents of parabolic equations and blow-up in R N. Proc. Roy. Soc. Edinburgh. Section A 128, 123–136 (1998)

    Article  MATH  Google Scholar 

  16. Seki Y.: On exact dead-core rates for a semilinear heat equation with strong absorption. Commun. Contemp. Math. 13, 1–52 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. C. C. Wu and Z. C. Zhang, Dead-core rates for the heat equation with a spatially dependent strong absorption, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), 2203–2210.

  18. Z. Q. Wu, J. N. Zhao, J. X. Yin, and H. L. Li, Nonlinear Diffusion Equations, World Scientific Publishing, River Edge, NJ, 2001.

  19. T. I. Zelenjak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Diff. Eqns 4 (1968), 17–22.

  20. Z. C. Zhang and B. Wang, Spatial Profile of the dead core for the fast diffusion equation with dependent coefficient, International J. Differential Equations Volume 2011, Article ID 751969, 9 pages.

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Pan Zheng, Chunlai Mu & Iftikhar Ahmed

  2. College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China

    Pan Zheng

Authors
  1. Pan Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunlai Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Iftikhar Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pan Zheng.

Additional information

This work is supported in part by NSF of China (11371384) and in part by the Fundamental Research Funds for the Central Universities, Project No. CDJXS 12 10 00 14.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zheng, P., Mu, C. & Ahmed, I. Dead-core rates for the fast diffusion equation with a spatially dependent strong absorption. Arch. Math. 102, 469–481 (2014). https://doi.org/10.1007/s00013-014-0643-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-014-0643-3

Mathematics Subject Classification (2000)

Keywords

Search

Navigation