Interaction between fast diffusion and geometry of domain

Abstract

Let $\Omega$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partial\Omega$ is not necessarily bounded. We consider two fast diffusion equations $\partial_t u= \mbox{div}(|\nabla u|^{p-2}{\nabla u})$ and $\partial_t u= \Delta u^{m}$, where $1<p<2$ and $0<m<1$. Let $u=u(x,t)$ be the solution of either the initial-boundary value problem over $\Omega$, 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 $\mathbb R^N\setminus \Omega$. Choose an open ball $B$ in $\Omega$ whose closure intersects $\partial\Omega$ 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 $u^\alpha$ over $B$ for short times in terms of principal curvatures of $\partial\Omega$ at the point, which tells us about the interaction between fast diffusion and geometry of domain.