On the local behavior of non-negative solutions to a logarithmically singular equation
type: Item
instance of: scholarly article
title: On the local behavior of non-negative solutions to a logarithmically singular equation
zbMATH Open document ID: 1255.35025
author: Emmanuele DiBenedetto
author: Ugo Gianazza
author: Naian Liao
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2012-09-12T00:00:00Z
review text: The authors study the local positivity of solutions to logarithmically singular diffusion equations. They extend recent intrinsic Harnack type proved for porous medium equations. More precisely it is shown that if at some time level \(t_0\in(0,T]\) and some point \(x_0\in E\) (where \(E\) is a space-time domain) the solution \(u(\cdot,t_0)\) is not identically zero in a neighborhood of \(x_0\), in a measure-theoretical sense, then it is strictly positive in a neighborhood of \((x_0, t_0)\) that is quantitatively determined. They are able to prove very interesting estimates for the porous medium equation \(u_t = \Delta (u^m)\) with \(m\) close to zero (\(m\) can be taken negative). To prove such a result, the authors use the so-called energy estimates, the DeGiorgi's approach and suitable \(L^1\)-\(L^1\) estimates.
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35K67
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6081725
zbMATH Keywords: logarithmic diffusion
zbMATH Keywords: expansion of positivity
zbMATH Keywords: Harnack-type estimates
zbMATH Keywords: intrinsic geometry
zbMATH Keywords: local positivity of solutions
zbMATH Keywords: porous medium equation
MaRDI profile type: Publication
full work available at URL: dcdsb.2012.17.1841
OpenAlex ID: W2323819119
DOI: 10.3934/DCDSB.2012.17.1841
description: scientific article; zbMATH DE number 6081725