Dynamics and Wong-Zakai approximations of stochastic nonlocal PDEs with long time memory

TL;DR

This paper studies stochastic nonlocal PDEs with long-term memory, proving solution existence, attractor properties, and analyzing Wong-Zakai approximation convergence to understand their long-term behavior under randomness.

Contribution

It introduces a combined Galerkin and Dafermos' transformation approach for existence, establishes attractor existence, and analyzes Wong-Zakai approximation convergence for stochastic nonlocal PDEs with memory.

Findings

Existence and uniqueness of solutions proved.

Tempered random attractors established.

Wong-Zakai approximations converge as step size approaches zero.

Abstract

In this paper, a combination of Galerkin's method and Dafermos' transformation is first used to prove the existence and uniqueness of solutions for a class of stochastic nonlocal PDEs with long time memory driven by additive noise. Next, the existence of tempered random attractors for such equations is established in an appropriate space for the analysis of problems with delay and memory. Eventually, the convergence of solutions of Wong-Zakai approximations and upper semicontinuity of random attractors of the approximate random system, as the step sizes of approximations approach zero, are analyzed in a detailed way.