Semidiscretization and long-time asymptotics of nonlinear diffusion equations

José A. Carrillo, Marco Di Francesco, Maria P. Gualdani

Research output: Contribution to journalReview articleResearchpeer-review

7 Citations (Scopus)

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. © 2007 International Press.
Original languageEnglish
Pages (from-to)21-53
JournalCommunications in Mathematical Sciences
Issue numberSUPPL. 1
Publication statusPublished - 30 Apr 2007

Keywords

  • Long-time asymptotics
  • Mass transport methods
  • Nonlinear diffusion

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