Skip to main content
SpringerLink
Log in

Large time behaviour of solutions of the porous media equation with absorption

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the large time behaviour of nonnegative solutions of the Cauchy problemu t=Δu m −u p,u(x, 0)=φ(x). Specifically we study the influence of the rate of decay ofφ(x) for large |x|, and the competition between the diffusion and the absorption term.

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. N. D. Alikakos and R. Rostamian,On the uniformization of the solutions of the porous medium equation in R n , Isr. J. Math.47 (1984), 270–290.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. I. Barenblatt,On some unsteady motions in a liquid or a gas in a porous medium, Prikladnaja Matematika i Mechanika16 (1952), 67–78.

    MATH  MathSciNet  Google Scholar 

  3. Ph. Benilan, M. G. Crandall and M. Pierre,Solutions of the porous medium equation in R N under optimal conditions of initial values, Indiana Univ. Math. J.33 (1984), 51–87.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Bertsch, R. Kersner and L. A. Peletier,Positivity versus localization in degenerate diffusion equations, Nonlinear Anal. TMA9 (1985), 987–1008.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Bertsch, T. Nanbu and L. A. Peletier,Decay of solutions of a degenerate nonlinear diffusion equation, Nonlinear Anal.6 (1982), 539–554.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Brézis, L. A. Peletier and D. Terman,A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Anal., to appear.

  7. B. E. Dahlberg and C. E. Kenig,Nonnegative solutions of the porous medium equations, Commun. Partial Differ. Equ.9 (1984), 409–437.

    Article  MATH  MathSciNet  Google Scholar 

  8. E. DiBenedetto,Continuity of weak solutions to a general porous media equation, Indiana Univ. Math. J.32 (1983), 83–118.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Escobedo and O. Kavian,Asymptotic behaviour of positive solutions of a nonlinear heat equation, to appear.

  10. A. Friedman and S. Kamin,The asymptotic behaviour of gas in an n-dimensional porous medium, Trans. Am. Math. Soc.262 (1980), 551–563.

    Article  MATH  MathSciNet  Google Scholar 

  11. V. A. Galaktionov, S. P. Kurdjumov and A. A. Samarskii,On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation, Mat. Sb.126 (1985), 435–472 (in Russian).

    MathSciNet  Google Scholar 

  12. A. Gmira and L. Veron,Large time behaviour of solutions of a semilinear equation in R n , J. Differ. Equ.53 (1984), 258–276.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. S. Kalashnikov,The propagation of disturbances of nonlinear heat conduction with absorption, USSR Comp. Math. Math. Phys.14 (1974), 70–85.

    Article  Google Scholar 

  14. S. Kamin and L. A. Peletier,Source type solutions of degenerate diffusion equations with absorption, Isr. J. Math.50 (1985), 219–230.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Kamin and L. A. Peletier,Singular solutions of the heat equation with absorption, Proc. Am. Math. Soc.95 (1985), 205–210.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Kamin and L. A. Peletier,Large time behaviour of solutions of the heat equation with absorption, Ann. Scu. Norm. Sup. PisaXII (1985), 393–408.

    MathSciNet  Google Scholar 

  17. S. Kamin and M. Ughi, to appear.

  18. F. B. Weissler,Rapidly decaying solutions of an ODE with application to semilinear elliptic and parabolic PDE’s, to appear.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tel Aviv University, Israel

    S. Kamin

  2. Mathematical Institute, University of Leiden, The Netherlands

    L. A. Peletier

Authors
  1. S. Kamin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. A. Peletier
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kamin, S., Peletier, L.A. Large time behaviour of solutions of the porous media equation with absorption. Israel J. Math. 55, 129–146 (1986). https://doi.org/10.1007/BF02801989

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02801989

Keywords

Search

Navigation