Carrillo, José A.; di Francesco, Marco; Gualdani, Maria P. Semidiscretization and long-time asymptotics of nonlinear diffusion equations. (English) Zbl 1145.35028 Commun. Math. Sci. 1, Suppl., 21-53 (2007). The authors review some results conserning the long-time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of this nonlinear diffusion method is proposed and their numerical properties analyzed. Long-time asymptotic results by numerical simulation are demonstrated and several open problems are proposed. They show that for general nonlinear diffusion equation the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for euclidean Wasserstein distance. It can be proved that the family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities near zero. Reviewer: Qin Mengzhao (Beijing) Cited in 3 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 35K65 Degenerate parabolic equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35A35 Theoretical approximation in context of PDEs Keywords:nonlinear diffusion; long-time asymptotics; method of lines; mass transport methods PDFBibTeX XMLCite \textit{J. A. Carrillo} et al., Commun. Math. Sci. 1, 21--53 (2007; Zbl 1145.35028) Full Text: DOI