Pullback Attractors Research Articles

Pullback and uniform exponential attractors for non-autonomous Oregonator systems

Abstract We consider the long-time global dynamics of non-autonomous Oregonator systems. This system is a coupled system of three reaction-diffusion equations, that arises from the Belousov-Zhabotinskii reaction. We first present some sufficient conditions for the existence of pullback and uniform exponential attractors for non-autonomous dynamical system. Then, we apply abstract results to prove the existence of a pullback exponential attractor for Oregonator systems affected by time-dependent forces and a uniform exponential attractor for Oregonator systems driven by quasi-periodic external forces.

Existence and Regularity of Random Attractors for Stochastic Evolution Equations Driven by Rough Noise

AbstractThis work establishes the existence and regularity of random pullback attractors for parabolic partial differential equations with rough nonlinear multiplicative noise under natural assumptions on the coefficients. To this aim, we combine tools from rough path theory and random dynamical systems. An application is given by partial differential equations with rough boundary noise, for which flow transformations are not available.

Wong-Zakai approximations and random attractors for nonlocal stochastic Schrödinger lattice systems in weighted spaces

This paper deals with the long-term behavior of nonlocal Schrödinger lattice systems with multiplicative white noise and their Wong-Zakai approximate systems in weighted spaces. We first prove the existence and uniqueness of tempered pullback attractors for the original stochastic systems and the Wong-Zakai approximations. Then, we establish the upper semicontinuity of these attractors for the Wong-Zakai approximate systems as the step length of the Wiener shift approaches zero.

Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Time-dependent uniform upper semicontinuity of pullback attractors for non-autonomous delay dynamical systems: Theoretical results and applications

In this paper we provide general results on the uniform upper semicontinuity of pullback attractors with respect to the time parameter for non-autonomous delay dynamical systems. Namely, we establish a criteria in terms of the multi-index convergence of solutions for the delay system to the non-delay one, locally pointwise convergence and local controllability of pullback attractors. As an application, we prove the upper semicontinuity of pullback attractors for a non-autonomous delay reaction-diffusion equation to the corresponding nondelay one over any bounded time interval as the delay parameter tends to zero.

Far-from-equilibrium attractors with full relativistic Boltzmann approach in boost-invariant and non-boost-invariant systems

We study the universal behavior associated with a Relativistic Boltzmann Transport (RBT) approach with the full collision integral in 0+1D conformal systems. We show that all momentum moments of the distribution function exhibit universal behavior. Furthermore, the RBT approach allows to calculate the full distribution function, showing that an attractor behavior is present in both the longitudinal and transverse momentum dependence. We compare our results to the far-from-equilibrium attractors determined with other approaches, such as kinetic theory in Relaxation Time Approximation (RTA) and relativistic hydrodynamic theories, both in their viscous (DNMR) an anisotropic (aHydro) formulations, finding a very similar evolution, but an even faster thermalization in RBT for higher order moments. For the first time, we extended this analysis also to study the attractor behavior under a temperature-dependent viscosity η/s(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta /s(T)$$\\end{document}, accounting also for the rapid increase toward the hadronic phase. We find that a partial breaking of the scaling behavior with respect to τ/τeq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au /\ au _{eq}$$\\end{document} emerges only at T≈Tc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T \\approx T_c$$\\end{document} generating a transient deviation from attractors; interestingly this in realistic finite systems may occur around the freeze-out dynamics. Finally, we investigate for the first time results beyond the boost-invariant picture, finding that also in such a case the system evolves toward the universal attractor. In particular, we present the forward and pull-back attractors at different space-time rapidities including rapidity regions where initially the distribution function is even vanishing.

Lower Semicontinuity of Pullback Attractors for a Non-autonomous Coupled System of Strongly Damped Wave Equations

Lower Semicontinuity of Pullback Attractors for a Non-autonomous Coupled System of Strongly Damped Wave Equations

Cardinality and IOD-type continuity of pullback attractors for random nonlocal equations on unbounded domains

Cardinality and IOD-type continuity of pullback attractors for random nonlocal equations on unbounded domains

Non-Newtonian incompressible fluids with nonlinear shear tensor and hereditary conditions

We consider a mathematical model with delay for non-Newtonian incompressible fluids in a bounded domain. Existence of global weak solutions is proved under suitable regularity on the initial data and the forces. Conditions for uniqueness are also given, but in general the results are stated in a multi-valued framework. Suitable multi-valued dynamical systems are well-posed, using basically L2(Ω)n×L2(−h,0;L2(Ω)n) or C([−h,0];L2(Ω)n) norms. Then the existence of pullback attractors is ensured acting on several universes, some of them of fixed bounded sets and others of tempered type, depending on parameters related to an integrability condition of the force and the delay term. Finally, relationships between these families of attractors are also provided, improving the characterization of attraction with respect to previous results.

Random attractors for three-dimensional stochastic globally modified Navier–Stokes equations driven by additive noise on unbounded domains

Abstract This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 𝒪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball’s idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time.

Sufficient and necessary criteria for backward asymptotic autonomy of pullback attractors with applications to retarded sine-Gordon lattice systems

In this paper, we investigate the backward asymptotic autonomy of pullback attractors for asymptotically autonomous processes. Namely, time-components of the pullback attractors semi-converge to the global attractors of the corresponding limiting semigroups as the time-parameter goes to negative infinity. The present article is divided into two parts: theories and applications. In the theoretical part, we establish a sufficient and necessary criterion with respect to the backward asymptotic autonomy via backward compactness of pullback attractors. Moreover, this backward asymptotic autonomy is considered by the periodicity of pullback attractors. As for the applications part, we apply the abstract results to non-autonomous retarded sine-Gordon lattice systems. By backward uniform tail-estimates of solutions, we prove the existence of a pullback and global attractor for such lattice systems such that the backward asymptotic autonomy is satisfied. Furthermore, it is also fulfilled under the assumptions of the periodicity for the non-delay forcing and the convergence for processes.

Generalized $$\varphi $$-Pullback Attractors for Evolution Processes and Application to a Nonautonomous Wave Equation

Generalized $$\varphi $$-Pullback Attractors for Evolution Processes and Application to a Nonautonomous Wave Equation

Dynamics of a one-dimensional non-autonomous laminated beam

In this paper is analyzed the pullback dynamics of a laminated beam model subject to fractional Laplacian dissipation, nonlinear source terms and non autonomous external forces. For each gamma exponent of the Laplacian in the open interval with endpoints 0 and 1/2, the model is well-posedness and the evolution process associated to solutions of problem possesses a pullback attractor for a general basin of attraction. To conclude, we prove that the attractors are upper semicontinuous as γ tends to 0. The result is new and it is the first time when the non-autonomous laminated beam is studied.

Critical transitions for scalar nonautonomous systems with concave nonlinearities: some rigorous estimates

The global dynamics of a nonautonomous Carathéodory scalar ordinary differential equation x′=f(t,x) , given by a function f which is concave in x, is determined by the existence or absence of an attractor-repeller pair of hyperbolic solutions. This property, here extended to a very general setting, is the key point to classify the dynamics of an equation which is a transition between two nonautonomous asymptotic limiting equations, both with an attractor-repeller pair. The main focus of the paper is to get rigorous criteria guaranteeing tracking (i.e. connection between the attractors of the past and the future) or tipping (absence of connection) for the particular case of equations x′=f(t,x−Γ(t)) , where Γ is asymptotically constant. Some computer simulations show the accuracy of the obtained estimates, which provide a powerful way to determine the occurrence of critical transitions without relying on a numerical approximation of the (always existing) locally pullback attractor.

Residually continuous pullback attractors for stochastic discrete plate equations

For a nonautonomous stochastic discrete plate equation driven by multiplicative noise, we prove the unique existence of a pullback random attractor, which is a family of pullback‐attracting random compact sets parameterized by time and samples. We then establish the residual dense continuity of the pullback random attractor on the time‐sample plane with respect to the Hausdorff metric. Even without the Cantor continuum hypothesis, we show that the set of all points of continuity of the pullback random attractor is an uncountable set with the continuity cardinality. These results explain both nonexplosive and nonimplosive phenomena for the random plate lattice system.

Asymptotically autonomous stability of kernel sections for lattice plate equations with nonlinear damping

We develop the theory of kernel sections of non-autonomous dynamical systems. Under some sufficient conditions, we establish some abstract results on the asymptotically autonomous stability of kernel sections, which suggests that the time-section of a kernel section is asymptotically stable to a global attractor of the autonomous dynamical system. A lattice plate equation with nonlinear damping and non-autonomous forcing is considered as an application. It is shown that the time-section of kernel sections of the non-autonomous lattice plate equations are nonempty, uniformly compact, pullback attracting, and asymptotically stable to a global attractor of the autonomous lattice plate equations.

Multivalued dynamics of non-autonomous reaction–diffusion equation with nonlinear advection term

In this paper, we investigate a reaction–diffusion population model with a nonlinear advection term and a time-dependent force given by the equation ut−Δu+α→⋅∇up=f(u)+h(t)in(τ,∞)×Ω,subject to the boundary condition u=0 on (τ,∞)×∂Ω. Here, Ω⊂RN with N≥1 is a bounded domain with smooth boundary, τ∈R, α→=(α1,…,αN) is a given advective direction and p>1. The presence of the nonlinear advection term α→⋅∇up introduces technical difficulties in the analysis, leading to a scenario where the uniqueness of weak solutions cannot be guaranteed. Consequently, the equation generates a multi-valued nonautonomous dynamical system. In this context, we establish the existence of minimal pullback attractors, considering universes of bounded and tempered sets. Moreover, we explore the relationships between these pullback attractors. Finally, we prove the upper semicontinuity of pullback attractors with respect to the advective vector α→.

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

Continuity and pullback attractors for a semilinear heat equation on time-varying domains

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term f(cdot ) to satisfy int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{L^{2}},ds<infty for all tin mathbb{R}, we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in H^{1} topology and the usual (L^{2},L^{2}) pullback mathscr{D}_{lambda}-attractor indeed can attract in the H^{1}-norm, provided that int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{H^{-1}(mathcal{O}_{s})},ds< infty and fin L^{2}_{mathrm{loc}}(mathbb{R},L^{2}(mathcal{O}_{s})).

Long-time behavior for evolution processes associated with non-autonomous nonlinear Schrödinger equation

We consider non-autonomous nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in a bounded smooth domain and time-dependent forcing that models the motion of waves in a quantum-mechanical system. We address the problem of the local and global well posedness and using rescaling of time we prove the existence of a compact pullback attractor for the associated evolution process. Moreover, we prove that the pullback attractor has finite fractal dimension. To the best of our knowledge, our approach has not been used in the literature to treat the non-autonomous Schrödinger equation.