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: http://dx.doi.org/10.4310/CMS.2007.v5.n5.a4

Authors

José A. Carrillo

Marco Di Francesco

Maria P. Gualdani

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

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