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Weak mean random attractors for non-local random and stochastic reaction-diffusion equations

Rubén Caballero, Pedro Marín-Rubio, José Valero

Abstract

In this paper, we prove the existence of weak pullback mean random attractors for a non-local stochastic reaction-diffusion equation with a nonlinear multiplicative noise. Also, we establish the existence and uniqueness of solutions and weak pullback mean random attractors for a deterministic nonlocal reaction-diffusion equations with random initial data.

Weak mean random attractors for non-local random and stochastic reaction-diffusion equations

Abstract

In this paper, we prove the existence of weak pullback mean random attractors for a non-local stochastic reaction-diffusion equation with a nonlinear multiplicative noise. Also, we establish the existence and uniqueness of solutions and weak pullback mean random attractors for a deterministic nonlocal reaction-diffusion equations with random initial data.
Paper Structure (6 sections, 27 theorems, 155 equations)

This paper contains 6 sections, 27 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: abstract theory of weak pullback mean random attractors
  3. Weak pullback mean attractors over probability spaces
  4. Weak pullback mean attractors over filtered probability spaces
  5. The mean random attractor for a non-local problem with random initial data
  6. The mean random attractor for a stochastic non-local problem

Key Result

Theorem 4

Wang Let $X$ be reflexive. Assume that $\mathcal{D}$ is an inclusion-closed collection. If the mean random dynamical system $\Phi$ possesses a weakly compact $\mathcal{D}$-pullback absorbing family $K=\{K(t)\}_{t\in\mathbb{R}}\in\mathcal{D}$, then $\Phi$ has a unique weak $\mathcal{D}$-pullback mean where $\overline{C}^{w}$ means the closure of $C$ in the weak topology of $L^{p}(\Omega,X)$.

Theorems & Definitions (38)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Definition 9
  • Definition 10
  • ...and 28 more