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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]

In this paper we study the fractal dimension of global attractors for a class of wave equations with (single-point) degenerate nonlocal damping. Both the equation and its linearization degenerate into linear wave equations at the degenerate point, and the usual approaches to calculate the dimension of the entirety of attractors do not work directly. Instead, we develop a new process concerning the dimension near the degenerate point individually and show the finite dimensionality of the attractor. [ABSTRACT FROM AUTHOR]

In this paper, we consider a three-dimensional nonlinear system consisting of the incompressible Navier-Stokes equations governing fluid motion coupled with the incompressible Maxwell's equations governing the magnetic field. This system is subject to a no-slip boundary condition for the (average) velocity and the perfectly conducting wall condition for the magnetic field. We prove some regularity properties of solutions of this system through an approximation procedure based on Galerkin approximation scheme. As a consequence, we are able to derive some suitable uniform estimates which allow us to show smoothness of the global attractor. Finally, we discuss the relationship between global attractors, invariant measures, time-average measures, and statistical solutions of the three-dimensional system of globally modified magnetohydrodynamics equations in the case of temporally independent forcing terms. [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]

This paper targets the long-time dynamics of a semilinear system modeling a binary mixture of solids with frictional dissipation mechanism acting only on the elastic equation and subjected to nonlinear source term. The coupling gives new contributions to the theory associated with nonlinear dynamics of partially damped semilinear systems of a binary mixture of solids. Using the quasi-stability methods, we prove the existence of a smooth finite-dimensional global attractor irrespective of the wave speeds of the system. Moreover, we establish the existence of exponential attractors. [ABSTRACT FROM AUTHOR]

We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of N ≥ 2 components occupying a bounded region Ω ⊂ R d , d ≤ 3 . Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if d ≤ 2 , we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if d = 3 , we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If d = 3 , this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application. [ABSTRACT FROM AUTHOR]

In this paper, observability inequalities and their influence on the dynamics of the solutions of nonlinear von Kármán beam are studied under nonlinear damping and external forces. The nonlinear von Kármán beam equations describe the nonlinear oscillations with large displacements. The nonlinear terms of the considered model pose new mathematical and technical difficulties in our analysis. Under quite general assumptions on nonlinear sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the considered problem. The main objective is to construct, from observability inequalities, a smooth global attractor and a generalized exponential attractor with finite fractal dimensions. This paper is the first to study the dynamics of the von Kármán beam from observability inequalities. The results obtained generalize and improve some previous results and can also be used in control theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlinear plate equation modeling the oscillations of suspension bridges. Under mixed boundary conditions consisting of simply supported and free boundary conditions, we obtain the global well-posedness of solutions in suitable function spaces. In addition, we use the perturbed energy method to prove the existence of a bounded absorbing set and establish a stabilizability estimate. Then, we derive the existence of a global attractor by verifying the asymptotic smoothness of the corresponding dissipative dynamical system.

Two one-parametric bifurcation problems for scalar nonautonomous ordinary differential equations are analyzed assuming the coercivity of the time-dependent function determining the equation and the concavity of its derivative with respect to the state variable. The skewproduct formalism leads to the analysis of the number and properties of the minimal sets and of the shape of the global attractor, whose abrupt variations determine the occurrence of local saddle-node, local transcritical and global pitchfork bifurcation points of minimal sets and of discontinuity points of the global attractor. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlinear plate equation modeling the oscillations of suspension bridges. Under mixed boundary conditions consisting of simply supported and free boundary conditions, we obtain the global well-posedness of solutions in suitable function spaces. In addition, we use the perturbed energy method to prove the existence of a bounded absorbing set and establish a stabilizability estimate. Then, we derive the existence of a global attractor by verifying the asymptotic smoothness of the corresponding dissipative dynamical system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of global attractors for a new semilinear Timoshenko–Ehrenfest type system. Firstly we establish the well-posedness of the system using Faedo–Galerkin method. By considering only one damping term acting on the vertical displacement, we prove the existence of a smooth finite dimensional global attractor using the recent quasi-stability theory. Our results holds for any parameters of the system. [ABSTRACT FROM AUTHOR]

This paper deals with a fully-dynamic piezoelectric beam model with a nonlinear force acting on the longitudinal displacements of the beam. The equations of motion follows a system of non-compactly coupled wave equation. As the magnetic effects are discarded by setting the magnetic permeability $ \mu $ to zero, the equations are fully decoupled to a single wave equation (electrostatic/quasi-static beam model). The existence of smooth finite-dimensional global attractors is proved by considering a damping injection to only one of the equations. Next, the existence of a finite set of determining functionals for the long-time behavior of the system is proved. Finally, to transition from the fully-dynamic beam model to the commonly-used electrostatic/quasi-static beam model, a singular limit problem $ \mu \to 0 $ is considered. This analysis allows to compare the attractors from one model to another in the sense of the upper semi-continuity of the attractor $ \mathcal{A}_{\mu} $ as $ \mu \to 0 $. [ABSTRACT FROM AUTHOR]

In this paper, a three-layer Rao–Nakra sandwich beam is considered where the core viscoelastic layer is constrained by the purely elastic or piezoelectric outer layers. In the model, uniform bending motions of the overall laminate are coupled to the longitudinal motions of the outer layers, and the shear of the middle layer contributes to the overall motion. Together with nonlinear damping injection and nonlinear source terms, the existence and uniqueness of local and global weak solutions are obtained by the nonlinear semigroup theory and the theory of monotone operators. The global existence of potential well solutions and the uniform energy decay rates of such a solution, given as a solution to a certain nonlinear ODE, are shown are proved under certain assumptions of the parameters and by the Nehari manifold. Finally, the existence of a smooth global attractor with finite fractal dimension, which is characterized as an unstable manifold of the set of stationary solutions, and exponential attractors for the associated dynamical system are proved. The present paper extends the linear analysis of the stability of the Rao–Nakra sandwich beam to nonlinear analysis in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of weighted equations with memory ∂u/∂t - div(a(x)∇u) - ... k(s)Δu(t - 1) ds + f(u) = g(x). The main characteristics of the model are that the equation contains a degenerated term with the weighted coefficient a(x) and a memory term. Under suitable assumptions on the weight function a(x), we first investigate the existence and uniqueness of weak solutions to the initial-boundary-value problem for the above-mentioned equation. Next, we prove the existence of a global attractor. The results in this paper will extend some results in [14, 16, 17] to the memory cases which have not been studied before. [ABSTRACT FROM AUTHOR]

Рассматриваются два варианта обобщенного нелокального уравнения Гинзбурга-Ландау. Оба эти варианта изучаются вместе с однородными краевыми условиями Дирихле. Для соответствующих начально-краевых задач показано существование решений при всех положительных значениях эволюционной переменной. Для решений начально-краевых задач получены явные формулы в виде рядов Фурье. Изучены свойстварешений соответствующих начально-краевых задач. Во второй части работы рассмотрен вопрос о существовании глобальных аттракторов для решений изучаемых краевых задач. Изучен вопрос о свойствах глобальных аттракторов. В частности, дан ответ о евклидовой размерности таких аттракторов.Приведены достаточные условия, при которых глобальный аттрактор будет конечномерным. Выделен вариант нелокального уравнения Гинзбурга-Ландау, когда глобальный аттрактор будет бесконечномерным.

In this work, we consider a model of functional differential variational inequalities formulated by a delay differential inclusion and an elliptic variational inequality in Banach spaces. We prove the global solvability of the problem and we have given some sufficient conditions to ensure the existence of decay solutions. Moreover, the existence of a global attractor for the semiflow governed by our system is presented thanks to the framework of Melnik and Valero [On attractors of multivalued semi-flows and differential inclusions. Set-Valued Anal. 1998;6:83–111]. The application to partial differential inclusions driven by obstacle constraints is given to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

Active–dissipative evolution equations emerge in a variety of physical and technological applications including liquid film flows, flame propagation, epitaxial film growth in materials manufacturing, to mention a few. They are characterized by three main ingredients: a term producing growth (active), a term providing damping at short length scales (dissipative) and a nonlinear term that transfers energy between modes and crucially produces a nonlinear saturation. The manifestation of these three mechanisms can produce large-time spatiotemporal chaos as evidenced by the Kuramoto-Sivashinsky equation (negative diffusion, fourth-order dissipation and a Burgers nonlinearity), which is arguably the simplest partial differential equation to produce chaos. The exact form of the terms (and in particular their Fourier symbol) determines the type of attractors that the equations possess. The present study considers the spatial analyticity of solutions under the assumption that the equations possess a global attractor. In particular, we investigate the spatial analyticity of solutions of a class of one-dimensional evolutionary pseudo-differential equations with Burgers nonlinearity, which are periodic in space, thus generalizing the Kuramoto-Sivashinsky equation motivated by both applications and their fundamental mathematical properties. Analyticity is examined by utilizing a criterion involving the rate of growth of suitable norms of the |$n$| th spatial derivative of the solution, with respect to the spatial variable, as |$n$| tends to infinity. An estimate of the rate of growth of the |$n$| th spatial derivative is obtained by fine-tuning the spectral method, developed elsewhere. We prove that the solutions are analytic if |$\gamma $|⁠ , the order of dissipation of the pseudo-differential operator, is higher than one. We also present numerical evidence suggesting that this is optimal, i.e. if |$\gamma $| is not larger that one, then the solution is not in general analytic. Extensive numerical experiments are undertaken to confirm the analysis and also to compute the band of analyticity of solutions for a wide range of active–dissipative terms and large spatial periods that support chaotic solutions. These ideas can be applied to a wide class of active–dissipative–dispersive pseudo-differential equations. [ABSTRACT FROM AUTHOR]

In this paper, using the Gromov–Hausdorff distances between two global attractors (which may be in disjoint phase spaces) and two semi‐dynamical systems introduced by Lee et al. (2020), we consider the continuous dependence of the global attractors and the stability of the semi‐dynamical systems on global attractors induced by the Brinkman–Forchheimer equation under variation of the domain. The results of this paper improve on previous results, which can compare any two systems in different phase spaces without the process of "pull‐backing" the perturbed systems to the original domain. [ABSTRACT FROM AUTHOR]

The asymptotic analysis of solutions of the two dimensional convective Brinkman-Forchheimer (CBF) equations ∂ t u − μ Δ u + (u ⋅ ∇) u + α u + β | u | r − 1 u + ∇ p = f , ∇ ⋅ u = 0 , for r ∈ [1,3], in unbounded domains Ω is carried out in this work. If the forcing term f is in the space L 2 (Ω) , then we show that the global attractor for 2D CBF equations. defined in Poincaré domains and general unbounded domains is compact not only in the L 2 -norm but also in the ℍ 1 -norm. The enstrophy equation as well as the asymptotic compactness of the semigroup associated with the 2D CBF equations is exploited in the proofs. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in $ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $. We then prove that the solution semigroup has a unique global attractor in $ E $. We finally prove that this attractor can be approximated in terms of upper semicontinuity of $ E $ by a finite-dimensional global attractor of a $ 2(2n+1) $-dimensional truncation system as $ n $ goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices. [ABSTRACT FROM AUTHOR]

We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a large class of nonlinear reaction-diffusion equations comprising disturbed Chaffee–Infante equations, for example. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from ones used in the previous published literature on the long time behavior of heat equations with memory, which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case. [ABSTRACT FROM AUTHOR]

We consider a nonlinear coupled PDE model for a single piezoelectric beam retaining the electromagnetic effects and a long-range strain memory. Nonlinear source terms in both mechanical and electromagnetic equations and a viscous magnetic damping term in the electromagnetic equation are considered in the model. The mathematical analysis of this model is particularly needed for certain class of fully dynamic piezoelectric materials demonstrating a viscoelastic memory or creep. With an injection of magnetic damping, the structure of the dynamical system associated with the solutions of this system allows using the quasi-stability theory in order to obtain the existence of global and exponential attractors. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of global attractors, extreme stability, periodicity and asymptotically periodicity of solutions of the delayed population model with survival rate on isolated time scales given by $ x^{\Delta} (t) = \gamma(t) x(t) + \dfrac{x(d(t))}{\mu(t)}e^{r(t)\mu(t)\left(1 - \frac{x(d(t))}{\mu(t)}\right)}, \ \ t \in \mathbb T. $ We present many examples to illustrate our results, considering different time scales.

In this paper, we study the regularity properties of solutions to the Davey–Stewartson system. It is shown that for initial data in a Sobolev space, the nonlinear part of the solution flow resides in a smoother space than the initial data for all times. We also obtain that the Sobolev norm of the nonlinear part of the evolution grows at most polynomially. As an application of the smoothing estimate, we study the long-term dynamics of the forced and weakly damped Davey–Stewartson system. In this regard, we give a new proof for the existence and smoothness of the global attractors in the energy space. [ABSTRACT FROM AUTHOR]

We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms. [ABSTRACT FROM AUTHOR]

The averaging process between two-dimensional fractional Navier–Stokes equations driven by a singularly oscillating external force and the averaged equations corresponding to the limiting case are investigated. The uniform boundedness of the global attractors for a fractional Navier–Stokes equation with a singularly external force is established. Furthermore, these global attractors converge uniformly to the attractor of the averaged equations under suitable assumptions on the singularly external force, and the explicit convergence rate of the global attractors is guaranteed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity ff satisfying the polynomial growth of arbitrary p−1p-1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the (L2(Ω),Lp(Ω))\left({L}^{2}\left(\Omega ),{L}^{p}\left(\Omega ))-global attractors immediately; moreover, such an attractor can attract every bounded subset of L2(Ω){L}^{2}\left(\Omega ) with the Lp+δ{L}^{p+\delta }-norm for any δ∈[0,+∞)\delta \in \left[0,+\infty ).

The evolution of a determining form for the 2D Navier–Stokes equations (NSE) which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the initial trajectory and a chosen, fixed steady state, with a dynamical convexity parameter θ, which will be called the characteristic determining parameter. That is, we show a separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter θ which is governed by a one-dimensional ODE. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than O(τ- 1 / 2) , otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than O(τ- 1) , as τ→ ∞, where τ is the evolutionary variable in determining form. The one-dimensional ODE is also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. It is shown that the zeros of the scalar function that governs the dynamics of θ, which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE.

In this paper we study the long-time behaviour of a system consisting of two nonlinear wave equations under the action of three competing forces, damping forces, strong source and external force. It is of great interest to know how the relationship between these forces acts on the behaviour of the solutions of the system. In this sense, we investigate the well-posedness of system, as well as the existence of global and exponential attractors. In addition, we consider the upper semicontinuity of the global attractor when the coupling parameter of the system tends to zero. Once proved the existence of global solutions (in time), to obtain the existence of global and exponential attractors results, we prove that the dynamical system associated to solutions of the model is quasi-stable and gradient. [ABSTRACT FROM AUTHOR]

In this paper we study a nonlocal reaction–diffusion equation in which the diffusion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Thirdly, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories. [ABSTRACT FROM AUTHOR]

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension. [ABSTRACT FROM AUTHOR]

The three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt (Kelvin-Voight) fluids in bounded domains is considered in this work. We investigate the long-term dynamics of such viscoelastic fluid flow equations with "fading memory" (non-autonomous). We first establish the existence of an absorbing ball in appropriate spaces for the semigroup defined for the Kelvin-Voigt fluid flow equations of order one with "fading memory" (transformed autonomous coupled system). Then, we prove that the semigroup is asymptotically compact, and hence we establish the existence of a global attractor for the semigroup. We provide estimates for the number of determining modes for both asymptotic as well as for trajectories on the global attractor. Once the differentiability of the semigroup with respect to initial data is established, we show that the global attractor has finite Hausdorff as well as fractal dimensions. We also show the existence of an exponential attractor for the semigroup associated with the transformed (equivalent) autonomous Kelvin-Voigt fluid flow equations with "fading memory". Finally, we show that the semigroup has Ladyzhenskaya's squeezing property and hence is quasi-stable, which also implies the existence of global as well as exponential attractor having finite fractal dimension. [ABSTRACT FROM AUTHOR]