Interaction between nonlinear diffusion and geometry of domain

TL;DR

This paper investigates how nonlinear diffusion processes interact with the geometric shape of the domain, deriving short-time asymptotics and geometric characterizations related to the diffusion behavior.

Contribution

It provides new asymptotic estimates for nonlinear diffusion in domains with complex boundaries and characterizes stationary level surfaces using the sliding method.

Findings

Asymptotic estimates relate diffusion content to domain geometry

Characterization of hyperplanes via stationary level surfaces

Insights into nonlinear diffusion and domain geometry interaction

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 $\partial_t u= \Delta \phi(u)$. 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.