Topics on a Logarithmic Diffusion Equation

Abstract

In this thesis, we prove the existence of solutions to the Dirichlet problem for a logarithmic diffusion equation can be established when the boundary datum satisfies a certain condition. We also show that if the boundary datum vanishes on an open subset of the side boundary then solutions in general do not exist. We present several local regularity properties of solutions to the logarithmic diffusion equation under certain assumptions including a Harnack-type inequality, the local analyticity of solutions, and an $L^1_{loc}$-type Harnack inequality. We also use the Harnack-type inequality to establish a topology by which local solutions to the porous medium equations converge to solutions to the logarithmic diffusion equation. The conclusions are examined and discussed in a series of examples and counter-examples.