Interaction between fast diffusion and geometry of domain

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3 Citations (Scopus)

Abstract

Let Ω be a domain in RN, where N ≥ 2 and ∂Ω is not necessarily bounded. We consider two fast diffusion equations ∂tu = div(|∇u|p-2∇u) and ∂tu = Δum, where 1 < p < 2 and 0 < m < 1. 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 is a positive continuous function, or the Cauchy problem where the initial datum equals a nonnegative continuous function multiplied by the characteristic function of the set RN\Ω. Choose an open ball B in Ω whose closure intersects ∂Ω only at one point, and let α > (N+1)(2-p)/2p or α > (N+1)(1-m)/4. Then, we derive asymptotic estimates for the integral of uα over B for short times in terms of principal curvatures of ∂Ω at the point, which tells us about the interaction between fast diffusion and geometry of domain.

Original languageEnglish
Pages (from-to)680-701
Number of pages22
JournalKodai Mathematical Journal
Volume37
Issue number3
DOIs
Publication statusPublished - 2014

Keywords

  • Cauchy problem
  • Fast diffusion
  • Geometry of domain
  • Initial behavior
  • Initial-boundary value problem
  • P-Laplacian
  • Porous medium type
  • Principal curvatures

ASJC Scopus subject areas

  • Mathematics(all)

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