Boundary Value Problems

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

Boundary Value Problems20062007:021425

https://doi.org/10.1155/2007/21425

©  M. Bonforte and J. L. Vazquez 2007

Received: 30 June 2006

Accepted: 20 September 2006

Published: 30 November 2006

Abstract

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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Authors’ Affiliations

(1)
Departamento de Matemáticas, Universidad Autónoma de Madrid
(2)
Centre De Recherche en Mathématiques de la Décision, Université Paris Dauphine

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Copyright

© M. Bonforte and J. L. Vazquez 2007

This article is published under license to BioMed Central Ltd. 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.