Pullback Exponential Attractors Research Articles

Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications

This article is a continuation of our previous work [5], where weformulated general existence theorems forpullback exponential attractors for asymptotically compact evolution processes inBanach spaces and discussed its implications in the autonomous case.We now study properties of the attractors anduse our theoretical results to prove the existence of pullback exponentialattractors in two examples, where previous results do not apply.

Dynamics of non-autonomous parabolic problems with subcritical and critical nonlinearities

This is a subsequent work of our previous one in [25]. Let the nonlinear terms of non-autonomous parabolic problems with singular initial data satisfy subcritical and critical growth conditions. We first establish the existence of uniform attractors in W1,r(Ω), 1<r<N, for the family of processes corresponding to the equations with external forces being translation bounded but not translation compact. Then, we prove the existence of pullback attractors in Lr(Ω) and W1,r(Ω), respectively, for the process corresponding to the equation with the weaker assumption on the external force than previous one. Finally, we investigate the robust of attractors and establish the existence of pullback exponential attractors for the process acting in Lr(Ω) and W1,r(Ω), respectively.

Pullback Exponential Attractors for Nonautonomous Klein-Gordon-Schrödinger Equations on Infinite Lattices

This paper proves the existence of the pullback exponential attractor for the process associated to the nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices.

Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results

We construct exponential pullback attractors for time continuous asymptotically compact evolution processes in Banach spaces and derive estimates on the fractal dimension of the attractors. We also discuss the corresponding results for autonomous processes.

Pullback Exponential Attractors for Non-autonomous Lattice Systems

We first present some sufficient conditions for the existence and the construction of a pullback exponential attractor for the continuous process (non-autonomous dynamical system) on Banach spaces and weighted spaces of infinite sequences. Then we apply our results to study the existence of pullback exponential attractors for first order non-autonomous differential equations and partly dissipative differential equations on infinite lattices with time-dependent coupled coefficients and time-dependent external terms in weighted spaces.

PULLBACK EXPONENTIAL ATTRACTORS FOR THE VISCOUS CAHN-HILLIARD EQUATION IN BOUNDED DOMAINS

This work is devoted to the construction of a pullback exponential attractor for a viscous Cahn-Hilliard system in bounded domains. Our con-struction is based on the results obtained by Langa, Miranville and Real in [7].

Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction–diffusion systems

The existence of a pullback exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction–diffusion equations with time-dependent external forces including the case of the FitzHugh–Nagumo system.

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 exponential attractors

In this work, we show how to construct a pullback exponentialattractor associated with an infinite dimensional dynamicalsystem, i.e., a family of time dependent compact sets, with finitefractal dimension, which are positively invariant andexponentially attract in the pullback sense every bounded set ofthe phase space. Our construction is based on the one in Efendiev etal. [11] in which a uniform forwards (and so alsopullback) exponential attractor is constructed. We relax theconditions in [11] in order to obtain an unbounded familyof exponential attractors for which the uniform convergence failsso that only the pullback attraction is expected. Thus, by provingthat global pullback attractors are included in our family ofexponential attractors, we generalize the concept of anexponential attractor to the theory of infinite dimensionalnon-autonomous dynamical systems. We illustrate our results on a 2DNavier-Stokes system in bounded domains.