Interaction between fast diffusion and geometry of domain

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 $1<p<2$ 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 $\mathbb R^N\setminus Ω$. Choose an open ball $B$ in $Ω$ whose closure intersects $\partialΩ$ 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 $u^α$ 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.

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