The aim of this notes is to include in a unitary presentation some topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued maximal monotone operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy.One case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsers of positive measure of the domain. Numerical simulations by which the theoretical results are applied to some concrete real-world problems are included with the double scope of verifying the theory and of illustrating the response given by the theoretical results to the problems arisen in applied sciences.