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

Abstract

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)(ω) =\text{Div} \left( g(ω) |DU(t)(ω)|^{p-2}DU(t)(ω) \right) \text{ on } S, g(ω)|DU(t)(ω)|^{p-2}DU(t)(ω)\cdotη=0 \text{ on } \partial S, U(0)(ω)=u(ω),\end{cases} \end{align} for $\mathbb{P}$-a.e. $\omega \in \Omega$ and a.e. $t\in (0,\infty )$ admits a unique strong solution and to determine asymptotic properties of this solution.