Asymptotic Results for Solutions of a weighted p-Laplacian evolution Equation with Neumann Boundary Conditions

Abstract

The purpose of this paper is to investigate the time behavior of the solution of a weighted $p$-Laplacian evolution equation, given by \begin{align} \label{eveq} \begin{cases} u_{t} = \text{div} \left(\gamma |\nabla u|^{p-2}\nabla u \right) & \text{on} (0,\infty)\times S, \\ \gamma|\nabla u|^{p-2}\nabla u\cdot\eta=0 & \text{on} (0,\infty)\times \partial S, \\ u(0,\cdot)=u_{0} & \text{on} S,\end{cases} \end{align} where $n \in \mathbb{N}\setminus \{1\}$, $p \in (1,\infty)\setminus \{2\}$, $S\subseteq \mathbb{R}^{n}$ is an open, bounded and connected set of class $C^{1}$, $\eta$ is the unit outer normal on $\partial S $, and $\gamma: S \rightarrow (0,\infty)$ is a bounded function which can be extended to an $A_{p}$-Muckenhoupt weight on $\mathbb{R}^{n}$. It will be proven that the solution converges in $L^{1}(S)$ to the average of the initial value $u_{0} \in L^{1}(S)$. Moreover, a conservation of mass principle, an extinction principle and a decay rate for the solution will be derived.