Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations
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
Boundary Value Problems 2007, 2007:021425 doi:10.1155/2007/21425Published: 30 November 2006
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.