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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 Bonforte12* 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.

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