Existence and nonexistence of solutions to a logarithmic diffusion equation in bounded domains

We construct solutions to the Dirichlet problem of a logarithmic diffusion equation with boundary value that could vanish somewhere. We also provide nonexistence results that show our existence theorems are, in some sense, optimal. Based on these results, we are able to construct a number of examples that complement our previous study of local behaviors of solutions to such an equation. Some preliminary results on the geometry of the vanishing set of local solutions are also reported.