Interaction between fast diffusion and geometry of domain

Kodai Mathematical Journal

Shigeru Sakaguchi¹ 

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

关键词: fast diffusion;    Cauchy problem;    initial-boundary value problem;    p-Laplacian;    porous medium type;    initial behavior;    principal curvatures;    geometry of domain;   

DOI  :  10.2996/kmj/1414674616

学科分类：数学（综合）

来源: Tokyo Institute of Technology, Department of Mathematics

【 摘 要 】

References(15)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 > $\frac {(N+1)(2-p)}{2p}$ or α > $\frac {(N+1)(1-m)}{4}$. 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.

【 授权许可】

Unknown