Pullback Exponential Attractors Research Articles

Stability of pullback exponential attractors for non-autonomous extensible suspension bridge equations

Stability of pullback exponential attractors for non-autonomous extensible suspension bridge equations

Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

Continuity Properties of Pullback and Pullback Exponential Attractors for Non-autonomous Plate with $$p-$$Laplacian

Continuity Properties of Pullback and Pullback Exponential Attractors for Non-autonomous Plate with $$p-$$Laplacian

Pullback dynamics of Lamé systems with time-dependent weak damping

We study the longtime dynamics of nonautonomous Lamé systems with time-dependent weak damping. Firstly we establish the existence of pullback exponential attractors. Then we study the existence of a minimal pullback attractor and its improved regularity. By assuming the damping mechanism is defined by a ɛ-family of coefficients we also explore the almost everywhere continuity of the attractor with respect to the parameter ɛ.

Pullback exponential attractors in nonlocal Mindlin's strain gradient porous elasticity

ABSTRACT In this paper, we study the long-time dynamics of a pullback exponential attractors for non-autonomous one dimensional system in nonlocal Mindlin's strain gradient porous elastic theory recently developed by Aouadi [Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticity. Z Angew Math Phys. 2022;73:111]. In this theory, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. Then we establish a Lipschitz stability result. We also prove the existence of pullback exponential attractors which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. Finally, we prove the upper-semicontinuity of minimal pullback attractors with respect to the perturbations parameter in natural energy space.

Existence and stability of pullback exponential attractors for a nonautonomous semilinear evolution equation of second order

We consider a nonautonomous semilinear evolution problem that models some sort of propagation problem in nonlinear elastic rods and nonlinear ion-acoustic waves. We investigate the existence and stability of a family of pullback exponential attractors for our problem under suitable growth and dissipativeness conditions. Moreover, we also prove the upper and lower semicontinuity of this family of pullback exponential attractors at time zero. As a particular case, we obtain the existence of the pullback attractor in an appropriate space, we prove its upper semicontinuity and, lastly, we obtain a regularity result of this pullback attractor.

Pullback exponential attractors for second-order lattice system with nonstandard growth condition

In this paper, we study the existence of pullback attractors and pullback exponential attractors for lattice dynamical system in time-dependent sequence space. First, we introduce a new sequence space with time-dependent variable exponents. Second, two abstract criteria (or sufficient conditions) about the existence of pullback attractors and pullback exponential attractors are established for infinite dimensional lattice dynamical systems on time-dependent spaces of infinite sequences. Finally, for making full use of the above-mentioned abstract criteria, we consider a second order lattice system with nonstandard growth nonlinearity, and then the existence of bi-space pullback attractors and pullback exponential attractors on a time-dependent Musielak–Orlicz space is obtained. In particular, we point out that these criteria and analytical skills can be utilized to deal with other lattice systems satisfying nonstandard growth conditions.

Well-posedness and dynamics of 2D Navier–Stokes equations with moving boundary

We investigate the longtime dynamical behavior of 2D Navier–Stokes equations with a moving boundary. We obtain the well-posedness and dissipation through the penalty method. Then, we derive the regularity by applying a new penalty. Finally, we show that the induced dynamical system has pullback exponential attractors.

Pullback exponential attractors for the three dimensional non-autonomous primitive equations of large scale ocean and atmosphere dynamics

Pullback exponential attractors for the three dimensional non-autonomous primitive equations of large scale ocean and atmosphere dynamics

TIME-DEPENDENT ASYMPTOTIC BEHAVIOR OF THE WAVE EQUATION WITH STRONG DAMPING ON <inline-formula><tex-math id="M1">$\mathbb{R}^{N}$</tex-math></inline-formula>

We study the longtime dynamics of non-autonomous wave equations with strong damping in the case of critical nonlinearity. First of all, when $1\leq p\leq p^{\ast}=\frac{N+2}{(N-2)_{+}}$, we get the well-posedness of strong damped equation with dime-dependent decay coefficient in $\mathcal{H}_{t}=H^{1}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$, and prove the quasi-stability of weak solution in $\mathcal{H}_{t,-1}=H^{1}(\mathbb{R}^{N})\times H^{-1}(\mathbb{R}^{N})$. Then the time-dependent attractor is proved in $\mathcal{H}_{t}$. Finally, by using the quasi-stability of weak solution, we establish the existence the pullback exponential attractor for non-autonomous dynamical system $(U(t,\tau),\mathcal{H}_{t},\mathcal{H}_{t,-1})$.

Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems

<p style='text-indent:20px;'>In this paper we study long-time behavior of first-order non-autono-mous lattice dynamical systems in square summable space of double-sided sequences using the cooperation between the discretized diffusion operator and the discretized reaction term. We obtain existence of a pullback global attractor and construct pullback exponential attractor applying the introduced notion of quasi-stability of the corresponding evolution process.</p>

Pullback exponential attractors for a class of non-Newtonian micropolar fluids

In this paper, we consider the pullback asymptotic behaviors of weak solutions for the 2D non-autonomous Non-Newtonian micropolar fluids. Employing the methods of ℓ-trajectories in [31], we first establish the existence of a finite dimensional pullback attractor and a pullback exponential attractor for the process {L(t,τ)}t≥τ associated with (1.2)-(1.4) in the phase space Xℓ. Then we construct a pullback exponential attractor for the process {U(t,τ)}t≥τ associated with (1.2)-(1.4) in the original phase space H×L2(Ω).

Pullback exponential attractors for some non‐autonomous dissipative dynamical systems

The main objective of this paper is to develop a mathematical framework of pullback exponential attractors for some non‐autonomous dissipative dynamical systems via the ℓ‐trajectories methods. As an application, we prove the existence of pullback exponential attractors for three‐dimensional non‐autonomous planetary geostrophic equations of large‐scale ocean circulation, which implies that the fractal dimension of each section of the pullback attractor is finite.

Pullback Exponential Attractors for Non-autonomous Recurrent Neural Networks with Discrete and Distributed Time-Varying Delays

First some sufficient conditions are presented for the existence and the construction of pullback exponential attractors for infinite dimensional non-autonomous dynamical systems with delays. This abstract result is then used to establish the existence of pullback exponential attractors for an infinite lattice model of non-autonomous recurrent neural networks with discrete and distributed time-varying delays.

Random pullback and uniform exponential attractors for stochastic non-autonomous Navier-Stokes equations

We first prove the existence of a random pullback exponential attractor for the 2D Navier-Stokes equation with additive noise and non-autonomous forcing. Then, we introduce the definition and existence conditions of a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system, and prove the existence of a random uniform exponential attractor for the 2D Navier-Stokes equation with additive noise and quasi-periodic external forces.

Pullback exponential attractors for differential equations with delay

We show the existence of an exponential attractor for non-autono-mous dynamical system with bounded delay. We considered the case of strong dissipativity then prove that the result remains for the weak dissipativity. We conclude then the existence of the global attractor and ensure the boundedness of its fractal dimension.

Global dynamics of nonautonomous Hindmarsh–Rose equations

Global dynamics of nonautonomous diffusive Hindmarsh–Rose equations on a three-dimensional bounded domain in neurodynamics is investigated. The existence of a pullback attractor is proved through uniform estimates showing the pullback dissipative property and the pullback asymptotical compactness. Then the existence of pullback exponential attractor is also established by proving the smoothing Lipschitz continuity in a long run of the solution process.

Robustness of Attractors for Non-autonomous Kirchhoff Wave Models with Strong Nonlinear Damping

The paper investigates the robustness of pullback attractors and pullback exponential attractors of the non-autonomous Kirchhoff wave models with strong nonlinear damping: $$u_{tt}- (1+\epsilon \Vert \nabla u\Vert ^2)\Delta u-\sigma (\Vert \nabla u\Vert ^2) \Delta u_t+f(u)=g(x,t)$$, where $$\epsilon \in [0,1]$$ is an extensibility parameter. It shows that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: $$1\le p<p^{**}\equiv \frac{N+4}{(N-4)^+}$$, (i) the related process has a pullback attractor $${\mathscr {A}}_\epsilon $$ in natural energy space $${\mathcal {H}}=(H^1_0\cap L^{p+1})\times L^2$$ for each $$\epsilon $$, and it is upper semicontinuous on the perturbation $$\epsilon $$; (ii) the related process has a partially strong pullback exponential attractor $${\mathcal {M}}_\epsilon $$ for each $$\epsilon $$, and it is Holder continuous on $$\epsilon \in [0,1]$$. These results deepen and extend those in recent literatures (Chueshov in J Diff Equ 252:1229–1262, 2012; Ding et al. in Appl Math Lett 76:40–45, 2018; Wang and Zhong in Discrete Contin Dyn Syst 7:3189–3209, 2013). The main novelty of this paper is that it provides a new method to investigate the upper semicontinuity of pullback attractors and the stability of pullback exponential attractors in supercritical nonlinearity case.

Robust attractors for a Kirchhoff-Boussinesq type equation

The paper studies the existence of the pullback attractors and robust pullback exponential attractors for a Kirchhoff-Boussinesq type equation: $ u_{tt}-\Delta u_{t}+\Delta^{2} u = div\Big\{\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Big\}+\Delta g(u)+f(x,t) $. We show that when the growth exponent $ p $ of the nonlinearity $ g(u) $ is up to the critical range: $ 1\leq p\leq p^*\equiv\frac{N+2}{(N-2)^{+}} $, (ⅰ) the IBVP of the equation is well-posed, and its solution has additionally global regularity when $ t&gt;\tau $; (ⅱ) the related dynamical process $ \{U_f(t,\tau)\} $ has a pullback attractor; (ⅲ) in particular, when $ 1\leq p&lt; p^* $, the process $ \{U_f(t,\tau)\} $ has a family of pullback exponential attractors, which is stable with respect to the perturbation $ f\in \Sigma $ (the sign space).

Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping

The main objective of this paper is to study the long-time behavior of solutions for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping for $ r&gt;4. $ Inspired by the the methods of $ \ell $-trajectories in [27], we will prove the existence of a finite dimensional pullback attractor and a pullback exponential attractor, which gives another way of considering the long-time behavior of the non-autonomous evolutionary equations.