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

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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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Correspondence to Matteo Bonforte.

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Bonforte, M., Vazquez, J.L. Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations. Bound Value Probl 2007, 021425 (2006) doi:10.1155/2007/21425

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Keywords

  • Cauchy Problem
  • Global Property
  • Dirichlet Condition
  • Harnack Inequality
  • Intermediate Time