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

Boundary Value Problems volume 2007, Article number: 021425 (2006) | Download Citation

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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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Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, Madrid, 28049, Spain

    • Matteo Bonforte
    •  & Juan Luis Vazquez

  2. Centre De Recherche en Mathématiques de la Décision, Université Paris Dauphine, Place de Lattre de Tassigny, Paris Cédex 16, 75775, France

    • Matteo Bonforte

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Correspondence to Matteo Bonforte.

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Keywords

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