Interaction between fast diffusion and geometry of domain

Abstract

Let $Ω$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partialΩ$ is not necessarily bounded. We consider two fast diffusion equations $\partial_t u= \mbox{div}(|\nabla u|^{p-2}{\nabla u})$ and $\partial_t u= Δu^{m}$, where $1 \frac {(N+1)(2-p)}{2p}$ 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 $\partialΩ$ at the point, which tells us about the interaction between fast diffusion and geometry of domain.