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

TL;DR

This paper proves the existence, uniqueness, and asymptotic behavior of solutions to a randomized weighted p-Laplacian evolution equation with Neumann boundary conditions.

Contribution

It establishes the well-posedness and long-term properties of solutions for a stochastic p-Laplacian evolution model with boundary conditions.

Findings

Existence and uniqueness of strong solutions

Asymptotic behavior characterized

Applicable to stochastic PDEs with Neumann 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)(\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. $\omega \in \Omega$ and a.e. $t \in (0,\infty)$ admits a unique strong solution and to determine asymptotic properties of this solution.