Existence, Uniqueness and Stability of Mild Solutions to a Stochastic Nonlocal Delayed Reaction–Diffusion Equation

The aim of this paper is to investigate the existence, uniqueness and stability of mild solutions to a stochastic delayed reaction–diffusion equation with spatial non-locality. This equation can be used to model the spatial–temporal evolution for age-structured spices perturbed by some random effects or the stochastic neural networks. The Banach fixed point theorem and a truncation method are adopted to establish the existence and uniqueness of mild solutions under both global and local Lipschitz conditions. Then, we explore the mean square exponential stability and almost sure exponential stability by employing the inequality techniques, the stochastic analysis techniques together with the properties of the nonlocal delayed term. Furthermore, we obtain the critical value of time delay τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} that guarantees the stability of the mild solutions. At last, our theoretic results are illustrated by application to the stochastic non-local delayed Nicholson blowflies equation with numerical simulations.