Archiv der Mathematik

, 97:353 | Cite as

Extinction for a fast diffusion equation with a nonlinear nonlocal source

Article

Abstract

In this article, the authors establish conditions for the extinction of solutions, in finite time, of the fast diffusion equation \({u_t=\Delta u^m+a\int_\Omega u^p(y,t)\,{d}y,\ 0 < m < 1,}\) in a bounded domain \({\Omega\subset R^N}\) with N > 2. More precisely speaking, it is shown that if p > m, any solution with small initial data vanishes in finite time, and if p < m, the maximal solution is positive in Ω for all t > 0. For the critical case p = m, whether the solutions vanish in finite time or not depends on the value of , where \({\mu=\int_{\Omega}\varphi(x)\,{d}x}\) and \({\varphi}\) is the unique positive solution of the elliptic problem \({-\Delta\varphi(x)=1,\ x\in \Omega; \varphi(x)=0,\ x\in\partial\Omega}\) .

Mathematics Subject Classification (2010)

Primary 35K55 Secondary 35K57 

Keywords

Fast diffusion equation Nonlocal source Extinction in finite time 

References

  1. 1.
    Anderson J.R.: Local existence and uniqueness of solutions of degenerate parabolic equations. Commun. Partial Differential Equations 16, 105–143 (1991)MATHCrossRefGoogle Scholar
  2. 2.
    Anderson J.R., Deng K.: Global existence for degenerate parabolic equations with a nonlocal forcing. Math. Meth. Appl. Sci. 20, 1069–1087 (1997)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Berryman J.G., Holland C.J.: Stability of the separable solution for fast diffusion. Arch. Ration. Mech. Anal. 74, 379–388 (1980)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Borelli M., Ughi M.: The fast diffusion equation with strong absorption: the instantaneous shrinking phenomenon. Rend. Istit. Mat. Univ. Trieste 26, 109–140 (1994)MathSciNetMATHGoogle Scholar
  5. 5.
    : Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources. J. Comput. Appl. Math. 202, 237–247 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Friedman A., Herrero M.A.: Extinction properties of semilinear heat equations with strong absorption. J. Math. Anal. Appl. 124, 530–546 (1987)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Friedman A., Kamin S.: The asymptotic behavior of gas in an N-dimensional porous medium. Trans. Amer. Math. Soc. 262, 551–563 (1980)MathSciNetMATHGoogle Scholar
  8. 8.
    Ferreira R., Vazquez J.L.: Extinction behavior for fast diffusion equations with absorption. Nonlinear Anal. 43, 943–985 (2001)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Galaktionov V.A., King J.R.: Fast diffusion equation with critical Sobolev exponent in a ball. Nonlinearity 15, 173–188 (2002)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Galaktionov V.A., Peletier L.A., Vazquez J.L.: Asymptotics of fast-diffusion equation with critical exponent, SIAM J. Math. Anal. 31, 1157–1174 (2000)MathSciNetMATHGoogle Scholar
  11. 11.
    Galaktionov V.A., Vazquez J.L.: Asymptotic behaviour of nonlinear parabolic equations with critical exponents A dynamical system approach. J. Funct. Anal. 100, 435–462 (1991)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Galaktionov V.A., Vazquez J.L.: Extinction for a quasilinear heat equation with absorption I. Technique of intersection comparison. Comm. Partial Differential Equations 19, 1075–1106 (1994)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Galaktionov V.A., Vazquez J.L.: Extinction for a quasilinear heat equation with absorption II A dynamical system approach. Comm. Partial Differential Equations 19, 1107–1137 (1994)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Gu Y.G.: Necessary and sufficient conditions of extinction of solution on parabolic equations. Acta. Math. Sinica 37, 73–79 (1994) (in Chinese)MathSciNetGoogle Scholar
  15. 15.
    Herrero M.A., Velazquez J.J.L.: Approaching an extinction point in one-dimensional semilinear heat equations with strong absorptions. J. Math. Anal. Appl. 170, 353–381 (1992)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kalashnikov A.S.: The nature of the propagation of perturbations in problems of non-linear heat conduction with absorption. USSR Comp. Math. Math. Phys. 14, 70–85 (1974)CrossRefGoogle Scholar
  17. 17.
    Kalashnikov A.S.: Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations. Uspekhi Mat. Nauk 42, 135–176 (1987)MathSciNetGoogle Scholar
  18. 18.
    Sabinina E.S.: On a class of nonlinear degenerate parabolic equations. Dolk. Akad. Nauk SSSR 143, 794–797 (1962)MathSciNetGoogle Scholar
  19. 19.
    Leoni G.: A very singular solution for the porous media equation u t = Δu mu p when 0 < m < 1. J. Differential Equations 132, 353–376 (1996)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Li Y.X., Wu J.C.: Extinction for fast diffusion equations with nonlinear sources. Electron J. Differential Equations 2005, 1–7 (2005)Google Scholar
  21. 21.
    Peletier L.A., Junning Z.: Large time behavior of solution of the porous media equation with absorption: The fast diffusion case. Nonlinear Anal. 14, 107–121 (1990)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Peletier L.A., Junning Z.: Source-type solutions of the porous media equation with absorption: The fast diffusion case. Nonlinear Anal. 17, 991–1009 (1991)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Sacks P.E.: Continuity of solutions of a singular parabolic equation. Nonlinear Anal. 7, 387–409 (1983)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Tian Y., Mu C.L.: Extinction and non-extinction for a p-Laplacian equation with nonlinear source. Nonlinear Anal. 69, 2422–2431 (2008)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Yin J.X., Li J., Jin C.H.: Non-extinction and critical exponent for a polytropic filtration equation. Nonl. Anal. 71, 347–357 (2009)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Yin J.X., Jin C.H.: Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources. Proc. Edinburgh Math. Soc. 52, 419–444 (2009)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Yin J.X., Jin C.H.: Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources. Math. Method. Appl. Sci. 30, 1147–1167 (2007)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Yuan H.J. et al.: Extinction and positive for the evolution p-Laplacian equation in R N. Nonl. Anal. TMA 60, 1085–1091 (2005)MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of MathematicsJilin UniversityChangchunPeople’s Republic of China

Personalised recommendations