  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.
