Long-term behavior of nonlocal reaction-diffusion equation under small random perturbations

TL;DR

This paper studies the long-term dynamics of a nonlocal reaction-diffusion equation under small random perturbations, demonstrating existence of random attractors and convergence to deterministic behavior as noise diminishes.

Contribution

It establishes the existence of random attractors for nonlocal reaction-diffusion equations driven by stationary noise and proves their upper semicontinuity under small stochastic perturbations.

Findings

Existence of random attractors for the stochastic nonlocal reaction-diffusion equation.

Convergence of solutions to the deterministic equation as noise parameters approach zero.

Upper semicontinuity of attractors with respect to perturbation parameters.

Abstract

In this paper, we investigate the nonlocal reaction-diffusion equation driven by stationary noise, which is a regular approximation to white noise and satisfies certain properties. We show the existence of random attractor for the equation. When stochastic nonlocal reaction-diffusion equation is driven by additive and multiplicative noise, we prove that the solution converges to the corresponding deterministic equation and establish the upper semicontinuity of the attractors as the perturbation parameter \delta and \epsilon both approaches zero.