Interaction between fast diffusion and geometry of domain

• Sakaguchi Shigeru

  Research Center for Pure and Applied Mathematics Graduate School of Information Sciences Tohoku University

Abstract

Let Ω be a domain in R^N, where N ≥ 2 and ∂Ω is not necessarily bounded. We consider two fast diffusion equations ∂_tu = div(|∇u|^p-2∇u) and ∂_tu = Δu^m, 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 R^N\Ω. Choose an open ball B in Ω whose closure intersects ∂Ω only at one point, and let α > $\frac {(N+1)(2-p)}{2p}$ or α > $\frac {(N+1)(1-m)}{4}$. Then, we derive asymptotic estimates for the integral of u^α 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.

Journal

• Kodai Mathematical Journal

  Kodai Mathematical Journal 37 (3), 680-701, 2014

  Department of Mathematics, Tokyo Institute of Technology

Keywords

• fast diffusion
• Cauchy problem
• initial-boundary value problem
• <i>p</i>-Laplacian
• porous medium type
• initial behavior
• principal curvatures
• geometry of domain

Details

• CRID: 1390001205271533696
• NII Article ID: 130004705919
• DOI: 10.2996/kmj/1414674616
• ISSN: 18815472
• ISSN: 03865991
• Text Lang: en
• Data Source:
  • JaLC
  • Crossref
  • CiNii Articles
  • KAKEN
• Abstract License Flag: Disallowed