Interaction between fast diffusion and geometry of domain
Shigeru Sakaguchi · Kodai Mathematical Journal · 2014
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 \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.