Global attractor for a 3D reaction-diffusion equation

Xiaosong Wang,Hongjun Wang,Lingrui Zhang

doi:10.12988/imf.2014.311221

Abstract

In this paper, we consider the long time behavior of a 3D reactiondiffusion equation and obtain the existence of global attractor for a autonomous reaction-diffusion system by establishing the asymptotical smoothness for the semigroup by decomposition method which introduced by Zelik (CPAA 2004). Mathematics Subject Classifications: 35B40, 35Q99, 80A22

Highlights

Let Ω ⊂ R3 be a bounded domain with sufficiently smooth boundary ∂Ω, we consider the non-autonomous 3D reaction-diffusion equation: ut − Δu + λu + u3 = f (x), x ∈ Ω, t ∈ R+, (1)
Since the global wellposedness and long time behavior of reaction-diffusion equation pay an important role in explain the stability of reaction diffusion phenomena in physics, biology, chemistry and engineering science, the reactiondiffusion equation has become a hot topic in pure and applied mathematical research
Chepyzhov and Vishik [1] gave a new method which is well suited to investigating equations arising in mathematical physics without unique solvability, the theory of trajectory attractor has developed and used to reaction-diffusion equation

Summary

Introduction

For the classical result of reaction-diffusion equation, such as the global existence of solutions in L2(Ω) and H01(Ω), long-time behavior for autonomous system (such as global attractor, Hausdorff dimension of attractor), we can refer to [9], [4], [3]. Yang and Sun [10] represented a general theory of norm-to-weak semigroup and applied to reaction-diffusion equation which obtained the existence of global attractor. We consider the long time behavior of a 3D reaction-diffusion equation and obtain the existence of global attractor for a autonomous reactiondiffusion system by establishing the asymptotical smoothness for the semigroup by decomposition method which introduced by Zelik [11] (CPAA 2004).

Existence of global absorbing set

Asymptotic smoothness for the semigroup