Contents Online
Communications in Mathematical Sciences
Volume 5 (2007)
Supp. 1
Supplemental Issue 1
Semidiscretization and Long-time Asymptotics of Nonlinear Diffusion Equations
Pages: 21 – 53
DOI: https://dx.doi.org/10.4310/CMS.2007.v5.n5.a4
Authors
Abstract
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.
Keywords
Nonlinear diffusion, long-time asymptotics, mass transport methods
2010 Mathematics Subject Classification
35B40, 35K55, 35K65
Published 1 January 2007