Israel Journal of Mathematics

, Volume 55, Issue 2, pp 129–146 | Cite as

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

  • S. Kamin
  • L. A. Peletier
Article

Abstract

We study the large time behaviour of nonnegative solutions of the Cauchy problemutumup,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.

References

  1. 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.MATHCrossRefMathSciNetGoogle Scholar
  2. 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.MATHMathSciNetGoogle Scholar
  3. 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.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Bertsch, R. Kersner and L. A. Peletier,Positivity versus localization in degenerate diffusion equations, Nonlinear Anal. TMA9 (1985), 987–1008.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Bertsch, T. Nanbu and L. A. Peletier,Decay of solutions of a degenerate nonlinear diffusion equation, Nonlinear Anal.6 (1982), 539–554.MATHCrossRefMathSciNetGoogle Scholar
  6. 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.Google Scholar
  7. 7.
    B. E. Dahlberg and C. E. Kenig,Nonnegative solutions of the porous medium equations, Commun. Partial Differ. Equ.9 (1984), 409–437.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    E. DiBenedetto,Continuity of weak solutions to a general porous media equation, Indiana Univ. Math. J.32 (1983), 83–118.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Escobedo and O. Kavian,Asymptotic behaviour of positive solutions of a nonlinear heat equation, to appear.Google Scholar
  10. 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.MATHCrossRefMathSciNetGoogle Scholar
  11. 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).MathSciNetGoogle Scholar
  12. 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.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    A. S. Kalashnikov,The propagation of disturbances of nonlinear heat conduction with absorption, USSR Comp. Math. Math. Phys.14 (1974), 70–85.CrossRefGoogle Scholar
  14. 14.
    S. Kamin and L. A. Peletier,Source type solutions of degenerate diffusion equations with absorption, Isr. J. Math.50 (1985), 219–230.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    S. Kamin and L. A. Peletier,Singular solutions of the heat equation with absorption, Proc. Am. Math. Soc.95 (1985), 205–210.MATHCrossRefMathSciNetGoogle Scholar
  16. 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.MathSciNetGoogle Scholar
  17. 17.
    S. Kamin and M. Ughi, to appear.Google Scholar
  18. 18.
    F. B. Weissler,Rapidly decaying solutions of an ODE with application to semilinear elliptic and parabolic PDE’s, to appear.Google Scholar

Copyright information

© Hebrew University 1986

Authors and Affiliations

  • S. Kamin
    • 1
  • L. A. Peletier
    • 2
  1. 1.Department of MathematicsTel Aviv UniversityIsrael
  2. 2.Mathematical InstituteUniversity of LeidenThe Netherlands

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