Interaction between nonlinear diffusion and geometry of domain

Let $\Omega$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partial\Omega$ is not necessarily bounded. We consider nonlinear diffusion equations of the form $\partial_t u= \Delta \phi(u)$. 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 equals $1$, or the Cauchy problem where the initial data is the characteristic function of the set $\mathbb R^N\setminus \Omega$. We consider an open ball $B$ in $\Omega$ whose closure intersects $\partial\Omega$ only at one point, and we derive asymptotic estimates for the content of substance in $B$ for short times in terms of geometry of $\Omega$. Also, we obtain a characterization of the hyperplane involving a stationary level surface of $u$ by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.