Interaction between fast diffusion and geometry of domain

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