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In this work, we carry out the asymptotic analysis of two- and three-dimensional convective Brinkman–Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and unbounded Poincaré domains (using asymptotic compactness property). In Poincaré domains, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors with respect to domains for CBF equations. We consider an expanding sequence of simply connected, bounded and smooth subdomains Ωm of the Poincaré domain Ω such that Ωm → Ω as m →∞. If 풜m and 풜 are the global attractors of CBF equations corresponding to Ωm and Ω, respectively, then we show that for large enough m, the global attractor 풜m enters into any neighborhood 풰(풜) of 풜. The presence of the Darcy term in CBF equations helps us to obtain the above mentioned results in general unbounded domains also. Finally, we discuss the quasi-stability property of the semigroup associated with CBF equations in bounded domains and establish the existence of finite fractal dimensional global as well as exponential attractors. [ABSTRACT FROM AUTHOR]

Existence of global attractors for a structural-acoustic system, subject to restricted boundary dissipation, is considered. Dynamics of the acoustic environment is given by a linear 3-D wave equation subject to locally distributed boundary dissipation, while the dynamics on the (flat) structural wall is given by a 2D-Kirchhoff-Boussinesq plate equation, subject to linear dissipation and supercritical nonlinear restoring forces. It is shown that the trajectories of the dynamical system defined on finite energy phase space are attracted asymptotically to a global attractor. The main challenges of the problem are related to: (i) superlinearity of the elastic energy of the structural component, (ii) Boussinesq effects of internal forces potentially leading to a finite time blowing up solutions, (iii) partially-restricted boundary dissipation placed on the interface only. The resulting system lacks dissipativity along with the suitable compactness properties, both corner stones of PDE dynamical system theories [1]. To contend with the difficulties, a new hybrid approach based on a suitable adaptation of the so called "energy methods" [2-3] and compensated compactness [4] has been developed. The geometry of the acoustic chamber plays a critical role. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of viscoelastic plate equations with past history. We first transform the original initial-boundary value problem into an equivalent one by means of the history space framework. Then we use the perturbed energy method to establish a stabilizability estimate. By employing the gradient property and quasi-stability of the dynamical system, we obtain the existence of a global attractor with finite fractal dimension [ABSTRACT FROM AUTHOR]

In this work, we consider the two-dimensional Oldroyd model for the non-Newtonian fluid flows (viscoelastic fluid) in Poincaré domains (bounded or unbounded) and study their asymptotic behavior. We establish the existence of a global attractor in Poincaré domains using asymptotic compactness property. Since the high regularity of solutions is not easy to establish, we prove the asymptotic compactness of the solution operator by applying Kuratowski’s measure of noncompactness, which relies on uniform-tail estimates and the flattening property of the solution. Finally, the estimates for the Hausdorff as well as fractal dimensions of global attractors are also obtained. [ABSTRACT FROM AUTHOR]

This study investigates the properties of solutions about one-dimensional wave equations connected in parallel under the effect of two nonlinear localized frictional damping mechanisms. First, under various growth conditions about the nonlinear dissipative effect, we try to establish the decay rate estimates by imposing minimal amount of support on the damping and provide some examples of exponential decay and polynomial decay. To achieve this, a proper observability inequality has been proposed and constructed based on some refined microlocal analysis. Then, the existence of a global attractor is proved when the damping terms are linearly bounded at infinity, a special weighting function has been used in this part, which eliminates undesirable terms of the higher order while contributing lower-order terms. Finally, we establish that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals.

We consider the hyperbolic relaxation of the viscous Cahn-Hilliard equation with a symport term. This equation is characterized by the presence of the additional inertial term τDΦtt that accounts for the relaxation of the diffusion flux. We suppose that τD is dominated by the viscosity coefficient δ. Endowing the equation with Dirichlet boundary conditions, we show that it generates a dissipative dynamical system acting on a certain phase-space, depending on τD. This system is shown to possess a global attractor that is upper semicontinuous at τD = δ = 0. Then, we construct a family of exponential attractors ℑτD,δ which is a robust perturbation of an exponential attractor of the Cahn-Hilliard equation; namely, the symmetric Hausdorff distance between ℑτD,δ and ℑ0,0 tends to 0 as (τD,δ) tends to (0,0) in an explicitly controlled way. Finally, we present numerical simulations of the time evolution of weak solutions as a function of parameters. [ABSTRACT FROM AUTHOR]

The initial boundary value problem for a class of fourth-order nonlinear wave equations with dissipative terms and dispersive terms is considered in this paper. We first establish the global existence and the exponential decay of the solution. Next, based on Condition (C) and semigroup theory arguments, the existence of the global attractor is derived. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long-time dynamical behavior of the MGT-Fourier system $\left\{ {\begin{array}{l} u_{ttt}+\alpha u_{tt}-\beta\Delta u_t-\gamma\Delta u+\eta\Delta\theta+f_1(u,u_t,\theta) = 0,\nonumber\\ \theta_t-\kappa\Delta\theta-\eta\Delta u_{tt}-\eta\alpha\Delta u_t+f_2(u,u_t,\theta) = 0.\nonumber \end{array}} \right. $ First we use the nonlinear semigroup theory to prove the well-posedness of the solutions. Then we establish the existence of smooth finite dimensional global attractors in the system by showing that the solution semigroup is gradient and quasi-stable. Furthermore, we investigate the existence of generalized exponential attractors.

This paper deals with the Gromov-Hausdor stability for systems generated of reaction-diffusion equations whose dffusion coecients are simultaneously large in a bounded smooth domains. The appropriated framework is presented to establish the conjugation between the attractors by means of ε-isometries. [ABSTRACT FROM AUTHOR]

In this paper, we propose the compact convergence approach to deal with the continuity of attractors of some reaction–diffusion equations under smooth perturbations of the domain subject to nonlinear Neumann boundary conditions. We define a family of invertible linear operators to compare the dynamics of perturbed and unperturbed problems in the same phase space. All continuity arising from small smooth perturbations will be estimated by a rate of convergence given by the domain variation in a C 1 topology. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Gromov–Hausdorff stability of global attractors for the 3D Navier–Stokes equations with damping under variations of the domain, which describes the complexity of the dynamics of the motion of a fluid flow. The Gromov–Hausdorff stability accounts for the Gromov–Hausdorff distance between two global attractors which may lie in disjoint phase spaces, as well as the stability of global attractors under perturbations of the domain. The same phase space cannot be used for the convergence via the Gromov–Hausdorff distance, which can be overcome, following Lee et al.(2020), by introducing a Banach space defined on a variable domain without "pull-backing" the perturbed system onto the original domain. [ABSTRACT FROM AUTHOR]

This paper deals with the long-time dynamics of a nonlinear system of coupled wave equations with fractional damping term and subjected to small perturbations of autonomous external forces. Inspired by the works of Chueshov and Lasiecka (J Dyn Differ Equ 16:469–512, 2004; Appl Math Optim Equ 58:195–241, 2008; Mem AMS 912:1–167, 2008; Von Karman evolution equations. Well-posedness and long time dynamics, Springer, New York, 2010) on the property of quasi-stability of dynamic systems, we show first the quasi-stability property holds for the problem considered here and then prove the existence of global and exponential attractors. We also prove the upper semicontinuity of global attractors with respect to the fractional exponent. Finally, we show the continuity of global attractors with respect to a pair of parameters in a residual dense set and their upper semicontinuities in a complete metric space. [ABSTRACT FROM AUTHOR]

A kind of nonlocal reaction-diffusion equations on an unbounded domain containing a fractional Laplacian operator is analyzed. To be precise, we prove the convergence of solutions of the equation governed by the fractional Laplacian to the solutions of the classical equation governed by the standard Laplacian, when the fractional parameter grows to 1. The existence of global attractors is investigated as well. The novelty of this paper is concerned with the convergence of solutions when the fractional parameter varies, which, as far as the authors are aware, seems to be the first result of this kind of problems in the literature. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the dynamics of one-dimensional monotone nonautonomous (cocycle) dynamical systems. A description of the structures of their invariant sets, omega limit sets, Bohr/Levitan almost periodic and almost automorphic motions, global attractors, pinched and minimal sets is given. An application of our general results is given to scalar differential and difference equations. [ABSTRACT FROM AUTHOR]