Interaction between fast diffusion and geometry of domain

2014/04/19 by Shigeru Sakaguchi, Sakaguchi, Shigeru
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1404.4915

23 pages, to appear in Kodai Math. J

openalex publication_date 2014/04/19 · arxiv created 2014/05/27 · arxiv updated 2014/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be a domain in \mathbb R^N, where N ≥ 2 and ∂Ω is not necessarily bounded. We consider two fast diffusion equations ∂_t u= div(|∇ u|^p-2∇ u) and ∂_t u= 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 \mathbb 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.

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