# Interaction between nonlinear diffusion and geometry of domain - Mathematics > Analysis of PDEs

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Abstract: Let $\Omega$ be a domain in $\mathbb R^N$, where $N \ge 2$ and$\partial\Omega$ is not necessarily bounded. We consider nonlinear diffusionequations of the form $\partial t u= \Delta \phiu$. Let $u=ux,t$ be thesolution of either the initial-boundary value problem over $\Omega$, where theinitial value equals zero and the boundary value equals 1, or the Cauchyproblem 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 thecontent of substance in $B$ for short times in terms of geometry of $\Omega$.Also, we obtain a characterization of the hyperplane involving a stationarylevel surface of $u$ by using the sliding method due to Berestycki, Caffarelli,and Nirenberg. These results tell us about interactions between nonlineardiffusion and geometry of domain.

Author: ** Rolando Magnanini, Shigeru Sakaguchi**

Source: https://arxiv.org/