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Title page for ETD etd-05202014-220122


Type of Document Dissertation
Author Liao, Naian
Author's Email Address naian.liao@vanderbilt.edu
URN etd-05202014-220122
Title Topics on a Logarithmic Diffusion Equation
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Emmanuele DiBenedetto Committee Chair
Anne Kenworthy Committee Member
Dechao Zheng Committee Member
Larry Schumaker Committee Member
Keywords
  • local behaviors
  • existence
  • singular equation
Date of Defense 2014-04-17
Availability unrestricted
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.

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