On the Local Behavior of Non-Negative Solutions to a Logarithmically Singular Equation

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.