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

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}$isanopen,boundedandconnectedsetofclass$C^{1}$,$\eta$istheunitouternormalon$\partial S $,and$\gamma: S \rightarrow (0,\infty)$isaboundedfunctionwhichcanbeextendedtoan$A_{p}$-Muckenhouptweighton$$\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.