Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations
- Matteo Bonforte1, 2Email author and
- Juan Luis Vazquez1
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.
Keywords
Authors’ Affiliations
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