Some applications of potential theory for a degenerate type diffusion equation

Abstract

This thesis is aimed to study boundary conditions for heat potentials for degeneratetype diffusion equations with initial condition and conductivity coefficient given by a time variable. Equations can be one-dimensional or multi-dimensional. The latter gives a new look to the problem. It is worth to mention that the coefficient is not always positive and therefore, it causes a degeneracy for the equation. The found boundary conditions make the solution, which is in the form of potential, unique. These boundary conditions are commonly called transparent boundary conditions or Kac’s boundary conditions. This kind of results first appeared in M. Kac’s work in the middle of last century. He developed potential theory and established further applications. Since then, many researchers have been studying potential theory related to the field. Some problems can be solved by simple integration, whereas others need more efforts to put in. Even though the works of past decades play an important role in potential theory, there exist problems which are still hard to solve. Hence, potential theory needs further developments and modern approaches.