Pullback attractors of nonautonomous reaction–diffusion equations

Abstract

In this paper, firstly we introduce the concept of norm-to-weak continuous cocycle in Banach space and give a technical method to verify this kind of continuity, then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle, using the measure of noncompactness. As an application, we prove the existence of pullback attractors in H 1 0 of the cocycle associated with the solutions for some nonlinear nonautonomous reaction-diffusion equations. The attractor pullback attracts all bounded subsets of H 1 0 in the norm of H 1 0 .

EXISTENCE OF PULLBACK ATTRACTOR FOR A REACTION–DIFFUSION EQUATION IN SOME UNBOUNDED DOMAINS WITH NON-AUTONOMOUS FORCING TERM IN H-1

“…For a bounded domain Ω, and a translation bounded function h ∈ L 2 loc (R; L 2 (Ω)), the existence of a uniform attractor in L p (Ω) is demonstrated in [Song & Zhong, 2008]. Finally, the reader can find similar results for several variants of our model in the references [Wang et al, 2007], [Prizzi, 2003], [Morillas & Valero, 2005], [Sun & Zhong, 2005], among others.…”

“…This, and the fact that the non-autonomous h belongs to the space L 2 loc (R; H −1 (Ω)), are the main novelties of our problem. The lack of compactness of the injection H 1 0 (Ω) ⊂ L 2 (Ω) (in the case of unbounded domains) implies that the standard techniques previously used, particularly the one involving the so-called flatenning property (see [Kloeden & Langa, 2007], [Li & Zhong, 2007], [Song & Wu, 2007], [Wang & Zhong, 2008], amongst others), which have been successfully used when Ω is bounded and h ∈ L 2 loc (R; L 2 (Ω)), do not work in our case.…”

“…the existence of a pullback attractor in the space H 1 0 (Ω) is proved in [Song & Wu, 2007], while in [Li & Zhong, 2007] the translation bounded condition (7) is weakened to…”

On the existence of pullback attractor for a two-dimensional shear flow with Tresca's boundary condition

“…First we recall the definition of a cocycle and a pullback attractor, then, following [10], recall the abstract theory, and in the end apply the latter to our problem. Definition 4.1.…”

Pullback Exponential Attractors for Nonautonomous Reaction Diffusion Equations in H<sub>0</sub><sup style="margin-left:-6px;">1</sup>

“…For the autonomous case, i.e., ( ) g t does not depend on the time, the asymptotic behaviors of the solution have been studied extensively in the framework of global attractor, see [4]- [6]. For the nonautonomous case, the asymptotic behaviors of the solution have been studied in the framework of pullback attractor, see [7]- [9]. Recently, the theory of pullback exponential attractor have been developed, see [1]- [3], and some methods are given to prove the existence of pullback exponential attractors.…”