Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Xiuling Gui, Jin Yang, Chunfeng Wang, Jing Hou, Ji Shu

TL;DR

The paper analyzes the long-term behavior of a nonlocal reaction-diffusion equation under small random perturbations by stationary noise. It first establishes the existence of random attractors for the stochastic nonlocal PDE and then proves convergence of solutions and upper semicontinuity of attractors to the corresponding deterministic system as the perturbation parameters vanish, handling both additive and multiplicative noise cases. The authors employ random dynamical systems techniques, including Ornstein–Uhlenbeck conjugations and random conjugate equations, to relate stochastic and deterministic attractors via Hausdorff distance. These results illuminate how small random perturbations affect the asymptotic structure of nonlocal PDEs and provide a rigorous link between stochastic and deterministic attractors in the zero-noise limit.

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 δand εboth approaches zero.

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

TL;DR

The paper analyzes the long-term behavior of a nonlocal reaction-diffusion equation under small random perturbations by stationary noise. It first establishes the existence of random attractors for the stochastic nonlocal PDE and then proves convergence of solutions and upper semicontinuity of attractors to the corresponding deterministic system as the perturbation parameters vanish, handling both additive and multiplicative noise cases. The authors employ random dynamical systems techniques, including Ornstein–Uhlenbeck conjugations and random conjugate equations, to relate stochastic and deterministic attractors via Hausdorff distance. These results illuminate how small random perturbations affect the asymptotic structure of nonlocal PDEs and provide a rigorous link between stochastic and deterministic attractors in the zero-noise limit.

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 δand εboth approaches zero.

Paper Structure

This paper contains 5 sections, 30 theorems, 305 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Attractors of nonlocal stochastic PDEs driven by stationary noise.
  4. Convergence of random attractors for stochastic nonlocal PDEs with additive noise
  5. Convergence of random attractors for stochastic nonlocal PDEs with multiplicative noise

Key Result

Theorem 3.1

Assume that function $a$ is globally Lipschitz and satisfies eq:2.1, $f \in C(\Bbb R )$ fulfills eq:2.2--eq:2.4, $h \in L_{loc}^2(\Bbb R^+; V^*)$ and $l \in L^2(\mathcal{O})$. Additionally, function $g$ satisfies eq:2.5--eq:2.6, and exist $\epsilon_0\in(0,1)$ such that for all $\epsilon\in(0,\epsilo

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 57 more