Boundary Value Problems 
Volume 2007 (2007), Article ID 21425, 31 pages
doi:10.1155/2007/21425
Research Article

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

Matteo Bonforte1,2 and Juan Luis Vazquez1

 1Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, Madrid 28049, Spain
2Centre De Recherche en Mathématiques de la Décision, Université Paris Dauphine, Place de Lattre de Tassigny, Paris Cédex 16 75775, France
Received 30 June 2006; Accepted 20 September 2006

Recommended by Vincenzo Vespri

Abstract

We investigate local and global properties of positive solutions to the fast diffusion equation ut=Δum in the good exponent range (d−2)+/d<m<1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space ℝd, 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 ℝd 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.