On the local behavior of non-negative solutions to a logarithmically singular equation

Emmanuele DiBenedetto; Ugo Gianazza; Naian Liao

doi:10.3934/dcdsb.2012.17.1841

Article Contents

2012, Volume 17, Issue 6: 1841-1858. Doi: 10.3934/dcdsb.2012.17.1841

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On the local behavior of non-negative solutions to a logarithmically singular equation

1.
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240
2.
Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia

Received: August 31, 2011

Revised: October 31, 2011

Published: August 2012

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Abstract

The local positivity of solutions to logarithmically singular diffusion equations is investigated in some open space-time domain $E\times(0,T]$. It is shown that if at some time level $t_o\in(0,T]$ and some point $x_o\in E$ the solution $u(\cdot,t_o)$ is not identically zero in a neighborhood of $x_o$, in a measure-theoretical sense, then it is strictly positive in a neighborhood of $(x_o, t_o)$. The precise form of this statement is by an intrinsic Harnack-type inequality, which also determines the size of such a neighborhood.

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Mathematics Subject Classification: Primary: 35K65, 35B65; Secondary: 35B45.

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