  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.
