Interaction between nonlinear 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 nonlinear diffusion equations of the form $\partial_t u= Δϕ(u)$. 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 equals 1, or the Cauchy problem where the initial data is the characteristic function of the set $\mathbb R^N\setminus Ω$. We consider an open ball $B$ in $Ω$ whose closure intersects $\partialΩ$ only at one point, and we derive asymptotic estimates for the content of substance in $B$ for short times in terms of geometry of $Ω$. 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.

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