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 $\alpha > \frac {(N+1)(2-p)}{2p}$ or $\alpha > \frac {(N+1)(1-m)}{4}$. 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.
Citation
Shigeru Sakaguchi. "Interaction between fast diffusion and geometry of domain." Kodai Math. J. 37 (3) 680 - 701, October 2014. https://doi.org/10.2996/kmj/1414674616
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