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

Reverse smoothing effects, fine asymptotics, and Harnack inequalities for fast diffusion equations

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.