On the local behavior of non-negative solutions to a logarithmically singular equation
Emmanuele DiBenedetto Ugo Gianazza Naian Liao
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.
keywords: logarithmic diffusion Singular parabolic equations expansion of positivity Harnack-type estimates.

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