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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.

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.

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 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.

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 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 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 ).

Abstract This paper deals with the asymptotic behavior of solutions to the initial-boundary value problem of the following fractional p-Kirchhoff equation: u t + M ( [ u ] s , p p ) ( − Δ ) p s u + f ( x , u ) = g ( x ) in Ω × ( 0 , ∞ ) , $$ u_{t}+M\bigl([u]_{s,p}^{p}\bigr) (-\Delta )_{p}^{s}u+f(x,u)=g(x)\quad \text{in } \Omega \times (0, \infty ), $$ where Ω ⊂ R N $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with Lipschitz boundary, N > p s $N>ps$ , 0 < s < 1 < p $0< s

Abstract In this paper, we prove that the existence of global attractors for a Kirchhoff wave equation with nonlinear damping and memory.

In this paper we prove the existence of uniform global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of non-autonomous complex fluids.

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.

We prove continuity of the flows and upper semicontinuity of global attractors for a Takeuchi-Yamada type equation with variable exponents.

In this work we obtain some continuity properties on the parameter q at p=q for the Takeuchi-Yamada problem which is a degenerate p-laplacian version of the Chafee-Infante problem. We prove the continuity of the flows and the equilibrium sets, and the upper semicontinuity of the global attractors.

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.

In this survey we provide an introduction to the theory of impulsive dynamical systems in both the autonomous and nonautonomous cases. In the former, we will show two different approaches which have been proposed to analyze such kind of dynamical systems which can experience some abrupt changes in their evolution (impulses). But, unlike the autonomous framework, the nonautonomous one is being developed right now and some progress is being obtained over the recent years. We will provide some results on how the theory of autonomous impulsive dynamical systems can be extended to cover such nonautonomous situations, which are more often to occur in the real world.

We investigate a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem generates a family of solution operators exhibiting dissipation and conservation. The solution operators admit a family of compact global attractors that are bounded in a more regular phase-space

We consider semilinear parabolic equations involving an operator that is X-elliptic with respect to a family of vector fields X with suitable properties. The vector fields determine the natural functional setting associated to the problem and the admissible growth of the non-linearity. We prove the global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity.

Informational Structures (IS) and Informational Fields (IF) have been recently introduced to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios for a system in any particular state. In this paper, we develop further steps in this direction, describing a proper continuous framework for an abstract formulation, which could serve as a prototype of the IIT postulates.

In a previous article [7], we proposed a model of phase separation in a binary mixture confined to a bounded region which may be contained within porous walls. The boundary conditions were derived from a mass conservation law and variational methods. In the present paper, we study the problem further. Using a Faedo-Galerkin method, we obtain the existence and uniqueness of a global solution to our problem, under more general assumptions than those in [7]. We then study its asymptotic behavior and prove the existence of an exponential attractor (and thus of a global attractor) with finite dimension.

In this paper we give two existence results for a class of degenerate diffusion equations with p-Laplacian. One is on a unique global strong solution, and the other is on a global attractor. It is also shown that the global attractor coincides with the unstable set of the set of all stationary solutions. As a by-product, an a-priori estimate for solutions of the corresponding elliptic equations is obtained.

It is shown that non-autonomous quasi-homogeneous dynamical systems admit a compact global attractor. The general results obtained here are applied to differential equations both in finite dimensional spaces and in infinite dimensional spaces, such as ordinary differential equations in Banach space and some types of evolutional partial differential equations.

This article is devoted to the study of linear periodic dynamical systems, possessing the property of uniform exponential stability. It is proved that if the Cauchy operator of these systems possesses a certain compactness property, then the asymptotic stability implies the uniform exponential stability. We also show applications to different classes of linear evolution equations, such as ordinary linear differential equations in the space of Banach, retarded and neutral functional differential equations, some classes of evolution partial differential equations.

This article is devoted to the study linear non-autonomous dynamical systems possessing the property of uniform exponential stability. We prove that if the Cauchy operator of these systems possesses a certain compactness property, then the uniform asymptotic stability implies the uniform exponential stability. For recurrent (almost periodic) systems this result is precised. We also show application for different classes of linear evolution equations: ordinary linear differential equations in a Banach space, retarded and neutral functional differential equations, and some classes of evolution partial differential equations.

For the linear equation $x'= A(t)x$ with recurrent (almost periodic) coefficients in an arbitrary Banach space, we prove that the asymptotic stability of the null solution and of all limit equations implies the uniform stability of the null solution.

We delineate a mathematical model for the dynamics of the spread of ectoparasites and the diseases transmitted by them. We present how the dynamics of the system depends on the three reproduction numbers belonging to three of the four possible equilibria and give a complete characterization of the structure of the global attractor in each possible case depending on the reproduction numbers.

We review recent results on global attractors of U(1)-invariant dispersive Hamiltonian systems. We study several models based on the Klein-Gordon equation and sketch the proof that in these models, under certain generic assumptions, the weak global attractor is represented by the set of all solitary waves. In general, the attractors may also contain multifrequency solitary waves; we give examples of systems which contain such solutions.