Interaction between nonlinear 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 nonlinear diffusion equations of the form ${\partial }_{t}u=\Delta \varphi (u)$. 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.

25 pages, no figures. Added some details to introduction. A couple of small changes. To appear in Journal Diff. Eqs

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