Pullback𝒟-Attractor of Nonautonomous Three-Component Reversible Gray-Scott System on Unbounded Domains

Anhui Gu

doi:10.1155/2013/719063

Abstract

The long time behavior of solutions of the nonautonomous three-components reversible Gray-Scott system defined on the entire spaceℝnis studied when the external forcing terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established inL2ℝn3andH1ℝn3, respectively. The pullback asymptotic compactness of solutions is proved by using uniform estimates on the tails of solutions on unbounded domains.

Highlights

In this paper, we consider the dynamical behavior of the nonautonomous three-components reversible Gray-Scott system ∂u ∂t =d1Δu − (F + k) u + u2V − Gu3 + Nw + f1 (t, x), ∂V ∂t d2ΔV − FV − u2V
The letters M is a generic positive constant which may change its value from line to line or even in the same line
Let D be a collection of families of subsets of X

Summary

Introduction

We consider the dynamical behavior of the nonautonomous three-components reversible Gray-Scott system. We will use the uniform estimates on the tails of solutions to circumvent the difficulty caused by the unboundedness of the domain. This idea was developed in [12] to prove the asymptotic compactness of solutions for autonomous parabolic equations on Rn and later extended to stochastic equations in, for example, [6, 13,14,15]. The letters M is a generic positive constant which may change its value from line to line or even in the same line

Preliminaries

Cocycle Related to Three-Component Reversible Gray-Scott System

Uniform Estimates of Solutions

Existence of Pullback Attractors