Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems

A family of compact and positively invariant sets with uniformly bounded fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the process is constructed. The existence of such a family, called a pullback exponential attractor, is proved for a nonautonomous semilinear abstract parabolic Cauchy problem. Specific examples will be presented in the forthcoming Part II of this work.     

Pullback attractors of nonautonomous reaction-diffusion equations

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 of the cocycle associated with the solutions for some nonlinear nonautonomous reaction-diffusion equations. The attractor pullback attracts all bounded subsets of in the norm of .     

Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains

In this paper, the existence of (L²(Rn),L²(Rn))-pullback attractors and (L²(Rn),H¹(Rn))-pullback attractors are proved for reaction-diffusion equation in unbounded domains.     

Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions

In this paper we prove the existence and uniqueness of a weak solution for a non-autonomous reaction-diffusion model with dynamical boundary conditions. After that, a continuous dependence result is established via an energy method, including in particular some compactness properties. Finally, the precedent results are used in order to ensure the existence of minimal pullback attractors in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition. The relation among these families is also discussed.     

Pullback attractors in nonautonomous difference equations

Nonautonomous difference equations are formulated as cocycles which generalize semigroups corresponding to autonomous difference equations. Pullback attractors are the appropriate generalization of autonomous attractors to cocycles. The existence of a pullback attractor follows when the difference equation cocycle has a pullback absorbing set. Results from the literature are outlined, including the construction of a Lyapunov function characterizing pullback attraction, and illustrated with several examples.     

Pullback exponential attractors for nonautonomous dynamical system in space of higher regularity

Under what condition, a process which exists a $(E,E)$-pullback exponential attractor implies the existence of $(E,V)$- pullback exponential attractor when $V$ embedded in $E$? We answer this question in this paper. As an application of this result, we prove the existence of pullback exponential attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity in $L^q(\Omega)(\forall q\geq 2)$ and $H_0^1(\Omega)$.     

Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction–diffusion equations

In this paper, we introduce the concept of norm-to-weak continuous process in a Banach space, and obtain the existence of pullback attractors for this kind of process. Then we give a new method for proving the existence of the pullback attractors. As an application, we obtain the existence of pullback attractors for nonautonomous reaction–diffusion equation in with exponential growth of the external force.     

Uniform attractors for impulsive reaction-diffusion equations

The goal of this paper is to consider the long time behavior of solutions of reaction-diffusion equations with impulsive effects at fixed moment of time. Under a new class of impulse function, we prove the existence of uniform attractors in the spaces and L^2p-2(Ω), respectively.     

Finite dimensionality of global attractors for a non-classical reaction-diffusion equation with memory

In this paper, we consider a periodic boundary value problem for a non-classical reaction-diffusion equation with memory. In other paper, we use the ω-limit compactness of the solution semigroup {S(t)}_t≥0 to get the existence of a global attractor. The main goal here is to give an estimate of the fractal dimension of the global attractor. By the fractal dimension theorem given by A.O. Celebi et al., we obtain that the fractal dimension of the global attractor for the problem is finite; this makes the results for the non-classical reaction-diffusion equations more substantial and perfect.     

Shadow systems and attractors in reaction-diffusion equations

For a pair of reaction diffusion equations with one diffusion coefficient very large, there is associated a reaction diffusion equation coupled with an ordinary differential equation (the shadow system) with nonlocal effects which has the property that it contains all of the essential dynamics of the original equations. Key words: Attractors, shadow systems, reaction-diffusion equations     

Random attractors of reaction-diffusion equations with multiplicative noise in L

In this paper, we study the random dynamical system (RDS) generated by the reaction-diffusion equation with multiplicative noise and prove the existence of a random attractor for such RDS in L^p(D) for any p?2.     

Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equations in H

In this paper, we prove the existence of the pullback attractor for the non-autonomous Benjamin-Bona-Mahony equations in H² by establishing the pullback uniformly asymptotical compactness.     

Pullback exponential attractors for the non-autonomous micropolar fluid flows

This paper investigates the pullback asymptotic behaviors for the non-autonomous micropolar fluid flows in 2D bounded domains. We use the energy method, combining with some important properties of the generated processes, to prove the existence of pullback exponential attractors and global pullback attractors and show that they both with finite fractal dimension. Further, we give the relationship between global pullback attractors and pullback exponential attractors.     

Exponential attractors for first-order lattice dynamical systems

In l², we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l² is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.     

Pullback attractors and invariant measures for the discrete Zakharov equations

This article studies the probability distributions of solutions in the phase space for the discrete Zakharov equations. The authors first prove that the generated process of the solutions operators possesses a pullback-${\mathcal D}$ attractor, and then they establish that there exists a unique family of invariant Borel probability measures supported by the pullback attractor.     

Uniform exponential attractor for nonautonomous partly dissipative lattice dynamical system

In this paper,we consider the existence of a uniform exponential attractor for the nonautonomous partly dissipative lattice dynamical system with quasiperiodic external forces.     

Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system

In this paper we construct a dynamical process (in general, multivalued) generated by the set of solutions of an optimal control problem for the three-dimensional Navier-Stokes system. We prove the existence of a pullback attractor for such multivalued process. Also, we establish the existence of a uniform global attractor containing the pullback attractor. Moreover, under the unproved assumption that strong globally defined solutions of the three-dimensional Navier-Stokes system exist, which guaranties the existence of a global attractor for the corresponding multivalued semiflow, we show that the pullback attractor of the process coincides with the global attractor of the semiflow.