A randomized weighted $p$-Laplacian evolution equation with Neumann boundary conditions

The purpose of this paper is to show that the randomized weighted $p$-Laplacian evolution equation given by \begin{align} \label{eveqrand} \begin{cases} U^{\prime}(t)(\omega) =$\text{Div}$ \left( g(\omega) |DU(t)(\omega)|^{p-2}DU(t)(\omega) \right) $\text{ on }$ S, g(\omega)|DU(t)(\omega)|^{p-2}DU(t)(\omega)\cdot\eta=0 $\text{ on }$ \partial S, U(0)(\omega)=u(\omega),\end{cases} \end{align} for $\mathbb{P}$-a.e. $\omega \in \Omega$anda\mathrm{.}e\mathrm{.}$t \in (0,\infty)$ admits a unique strong solution and to determine asymptotic properties of this solution.