We review several results concerning the long-time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of these nonlinear diffusion models are proposed and their numerical properties analyzed. We demonstrate the long-time asymptotic results by numerical simulation and we discuss several open problems based on these numerical results. We show that for general nonlinear diffusion equations the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for the euclidean Wasserstein distance. In fact, we propose a new scaling for which we can prove that this family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities near zero.
"Semidiscretization and Long-time Asymptotics of Nonlinear Diffusion Equations." Commun. Math. Sci. 5 (S1) 21 - 53, February 2007.