This article is part of the series Harnack's Estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations.

Open Access Research Article

Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

Matteo Bonforte1,2* and Juan Luis Vazquez1

Author Affiliations

1 Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, Madrid 28049, Spain

2 Centre De Recherche en Mathématiques de la Décision, Université Paris Dauphine, Place de Lattre de Tassigny, Paris Cédex 16 75775, France

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Boundary Value Problems 2007, 2007:021425 doi:10.1155/2007/21425


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/021425


Received:30 June 2006
Accepted:20 September 2006
Published:30 November 2006

© 2007 Bonforte and Vazquez

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate local and global properties of positive solutions to the fast diffusion equation in the good exponent range , corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space , we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; also a slight improvement of the intrinsic Harnack inequality is given. We use them to derive sharp global positivity estimates and a global Harnack principle. Consequences of these latter estimates in terms of fine asymptotics are shown. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak, intrinsic, and elliptic Harnack inequalities for intermediate times. We also prove elliptic Harnack inequalities near the extinction time, as a consequence of the study of the fine asymptotic behavior near the finite extinction time.

References

  1. Vazquez, JL: Asymptotic beahviour for the porous medium equation posed in the whole space. Journal of Evolution Equations. 3(1), 67–118 (2003). Publisher Full Text OpenURL

  2. Vazquez, JL: The Dirichlet problem for the porous medium equation in bounded domains. Asymptotic behavior. Monatshefte für Mathematik. 142(1-2), 81–111 (2004). PubMed Abstract | Publisher Full Text | PubMed Central Full Text OpenURL

  3. Vazquez, JL: Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Lecture Notes, Oxford University Press, New York (2006)

  4. Bonforte, M, Vazquez, JL: Fine asymptotics and elliptic Harnack inequalities near the extinction time for fast diffusion equation. preprint, 2006

  5. Bonforte, M, Vazquez, JL: Global positivity estimates and Harnack inequalities for the fast diffusion equation. preprint, 2005

  6. Chen, YZ, DiBenedetto, E: On the local behavior of solutions of singular parabolic equations. Archive for Rational Mechanics and Analysis. 103(4), 319–345 (1988)

  7. DiBenedetto, E, Kwong, YC, Vespri, V: Local space-analyticity of solutions of certain singular parabolic equations. Indiana University Mathematics Journal. 40(2), 741–765 (1991). Publisher Full Text OpenURL

  8. Aronson, DG, Caffarelli, LA: The initial trace of a solution of the porous medium equation. Transactions of the American Mathematical Society. 280(1), 351–366 (1983). Publisher Full Text OpenURL

  9. Herrero, MA, Pierre, M: The Cauchy problem for when . Transactions of the American Mathematical Society. 291(1), 145–158 (1985)

  10. Lee, K-A, Vazquez, JL: Geometrical properties of solutions of the porous medium equation for large times. Indiana University Mathematics Journal. 52(4), 991–1016 (2003)

  11. Carrillo, JA, Vazquez, JL: Fine asymptotics for fast diffusion equations. Communications in Partial Differential Equations. 28(5-6), 1023–1056 (2003). Publisher Full Text OpenURL

  12. Bénilan, P, Crandall, MG: Regularizing effects of homogeneous evolution equations. Contributions to Analysis and Geometry (Baltimore, Md., 1980), Suppl. Am. J. Math., pp. 23–39. Johns Hopkins University Press, Maryland (1981)

  13. Berryman, JG, Holland, CJ: Stability of the separable solution for fast diffusion. Archive for Rational Mechanics and Analysis. 74(4), 379–388 (1980)

  14. Adimurthi, Yadava, SL: An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem. Archive for Rational Mechanics and Analysis. 127(3), 219–229 (1994). Publisher Full Text OpenURL

  15. Ni, W-M, Nussbaum, RD: Uniqueness and nonuniqueness for positive radial solutions of . Communications on Pure and Applied Mathematics. 38(1), 67–108 (1985). Publisher Full Text OpenURL

  16. Budd, C, Norbury, J: Semilinear elliptic equations and supercritical growth. Journal of Differential Equations. 68(2), 169–197 (1987). Publisher Full Text OpenURL

  17. Galaktionov, VA, Vazquez, JL: A Stability Technique for Evolution Partial Differential Equations. A Dynamical System Approach, Progress in Nonlinear Differential Equations and Their Applications,p. xxii+377. Birkhäuser Boston, Massachusetts (2004)

  18. Hale, JK: Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs,p. x+198. American Mathematical Society, Rhode Island (1988)

  19. DiBenedetto, E, Kwong, YC: Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations. Transactions of the American Mathematical Society. 330(2), 783–811 (1992). Publisher Full Text OpenURL

  20. Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften,p. x+401. Springer, Berlin (1977)

  21. Aronson, DG, Peletier, LA: Large time behaviour of solutions of the porous medium equation in bounded domains. Journal of Differential Equations. 39(3), 378–412 (1981). Publisher Full Text OpenURL