Interaction between nonlinear diffusion and geometry of domain

Rolando Magnanini, Shigeru Sakaguchi

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

Let Ω be a domain in RN, where N≥2 and ∂Ω is not necessarily bounded. We consider nonlinear diffusion equations of the form ∂tu=δφ(u). Let u=u(x,t) be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set RN\Ω. We consider an open ball B in Ω whose closure intersects ∂ Ω only at one point, and we derive asymptotic estimates for the content of substance in B for short times in terms of geometry of Ω. Also, we obtain a characterization of the hyperplane involving a stationary level surface of u by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.

Original languageEnglish
Pages (from-to)236-257
Number of pages22
JournalJournal of Differential Equations
Volume252
Issue number1
DOIs
Publication statusPublished - 2012 Jan 1
Externally publishedYes

Keywords

  • Cauchy problem
  • Geometry of domain
  • Initial behavior
  • Initial-boundary value problem
  • Nonlinear diffusion

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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