Abstract: This chapter begins with a systematic study of the questions of existence, uniqueness, and main properties of the solutions of the PME by concentrating on the first boundary-value problem posed in a spatial domain , which is a bounded subdomain of ^d , d 1. It focuses on homogeneous Dirichlet boundary conditions, u = 0 on , in order to obtain a simple problem for which a fairly complete theory can be easily developed as a first stage in understanding the theory of the PME. This is called the homogeneous Cauchy-Dirichlet problem, or more simply, the homogeneous Dirichlet problem.

Keywords: PME, weak solution, Dirichlet problem, Cauchy-Dirichlet problem, fast diffusion,