Compact Attractor Research Articles

Global and exponential attractors for viscoelastic Kirchhoff plate equation with memory and time delay

In this paper, we consider the viscoelastic Kirchhoff plate equation with memory and time delay. Under appropriate conditions on the real numbers and , we prove the existence of a compact global attractor with finite fractal dimension through a stabilizability estimate. Additionally, we demonstrate the existence of a fractal exponential attractor. To the best of our knowledge, these findings are novel for for this system.

Threshold dynamics of a degenerated diffusive incubation period host–pathogen model with saturation incidence rate

In this paper, we consider an incubation period host–pathogen system with degenerated diffusion. The global compact attractor of the solution of the model is investigated using the κ-contraction method. Furthermore, the basic reproduction number is defined, and we discuss the dynamic analysis of a degenerated diffusion model. The obtained theoretical results are nontrivial and can be considered a continuation of the work by Wang et al. in 2022.

Asymptotic smoothness effects and global attractor for a peridynamic model with energy damping

AbstractIn this paper, we consider a peridynamic model with energy damping inspired by the works of Balakrishnan and Taylor on “damping models” based on the instantaneous total energy of the system. We study the asymptotic behavior of solutions, in the sense of attractors, of these peridynamic models in suitable phase space; more precisely, we prove a result of existence and characterization of compact global attractors with a nonlinear strongly continuous semigroup approach based in the asymptotic smoothness property thanks to Chueshov and Lasiecka and Nakao's lemma.

Global dynamics and threshold behavior of an SEIR epidemic model with nonlocal diffusion

This paper studies the global dynamics of an SEIR (Susceptible–Exposed–Infectious–Recovered) model with nonlocal diffusion. We show the model’s well-posedness, proving the solutions’ existence, uniqueness, and positivity, along with a disease-free equilibrium. Next, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number R0, defined as the spectral radius of the next-generation operator. We show that the solution map has a global compact attractor, offering insights into long-term dynamics. In particular, the analysis shows that for R0<1, the disease-free equilibrium is globally stable. Using the persistence theory, we show that there is an endemic equilibrium point for R0>1. Moreover, by constructing an appropriate Lyapunov function, we establish the global stability of the unique endemic equilibrium in two distinct scenarios.

Long time dynamics of nonequilibrium electroconvection

The Nernst-Planck-Stokes (NPS) system models electroconvection of ions in a fluid. We consider the system, for two oppositely charged ionic species, on three dimensional bounded domains with Dirichlet boundary conditions for the ionic concentrations (modelling ion selectivity), Dirichlet boundary conditions for the electrical potential (modelling an applied potential), and no-slip boundary conditions for the fluid velocity. In this paper, we obtain quantitative bounds on solutions of the NPS system in the long time limit, which we use to prove (1) the existence of a compact global attractor with finite fractal (box-counting) dimension and (2) space-time averaged electroneutrality ρ ≈ 0 \rho \approx 0 in the singular limit of Debye length going to zero, ϵ → 0 \epsilon \to 0 .

Qualitative analysis on a reaction–diffusion SIS epidemic model with nonlinear incidence and Dirichlet boundary

In this paper, the dynamical behavior in a spatially heterogeneous reaction–diffusion SIS epidemic model with general nonlinear incidence and Dirichlet boundary condition is investigated. The well-posedness of solutions, including the global existence, nonnegativity, ultimate boundedness, as well as the existence of compact global attractor, are first established, then the basic reproduction number R0 is calculated by defining the next generation operator. Secondly, the threshold dynamics of the model with respect to R0 are studied. That is, when R0<1 the disease-free steady state is globally asymptotically stable, and when R0>1 the model is uniformly persistent and admits one positive steady state, and under some additional conditions the uniqueness of positive steady state is obtained. Furthermore, some interesting properties of R0 are established, including the calculating formula of R0, the asymptotic profiles of R0 with respect to diffusion rate dI, and the monotonicity of R0 with diffusion rate dI and domain Ω. In addition, the bang–bang-type configuration optimization of R0 also is obtained. This rare result in diffusive equation reveals that we can control disease diffusion at least at one peak. Finally, the numerical examples and simulations are carried out to illustrate the rationality of open problems proposed in this paper, and explore the influence of spatial heterogeneous environment on the disease spread and make a comparison on dynamics between Dirichlet boundary condition and Neumann boundary condition.

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.

Dynamics of a class of extensible beams with degenerate and non-degenerate nonlocal damping

This work is concerned with new results on long-time dynamics of a class of hyperbolic evolution equations related to extensible beams with three distinguished nonlocal nonlinear damping terms. In the first possibly degenerate case, the results feature the existence of a family of compact global attractors and a thickness estimate for their Kolmogorov’s $\varepsilon$-entropy. Then, in the non-degenerate context, the structure of the helpful nonlocal damping leads to the existence of finite-dimensional compact global and exponential attractors. Lastly, in a degenerate and critical framework, it is proved the existence of a bounded closed global attractor but not compact. To the proofs, we provide several new technical results by means of refined estimates that open up perspectives for a new branch of nonlinearly damped problems.

Dynamics of a diffusive delayed viral infection model in a heterogeneous environment

This paper investigates the asymptotic analysis of spatially heterogeneous viral transmission, incorporating cell‐to‐cell transmission, virus nonlocal dispersal, and intracellular delay. Due to the noncompactness of the semiflow, we used the Kuratowski measure of noncompactness to demonstrate the existence of a global compact attractor. This noncompactness issue generates difficulties in calculating the basic reproduction number , which is the principal eigenvalue of the next‐generation operator. The threshold role of this number is determined, where we derived two different cases: (i) the global stability of the virus‐free steady state, which is globally stable for by the Lyapunov direct method, and (ii) the global stability of the virus steady state for . Indeed, the second case is demonstrated through several steps that include uniform persistence, the existence of a virus steady state, and the global stability of the virus steady state. The results are supported by different graphical representations with proper biological justifications.

Poisson Stable Motions and Global Attractors of Symmetric Monotone Nonautonomous Dynamical Systems

This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo-periodic, pseudo-recurrent and Poisson stable) motions of symmetric monotone non-autonomous dynamical systems (NDS). It is proved that every precompact motion of such system is asymptotically Poisson stable. We give also the description of the structure of compact global attractor for monotone NDS with symmetry. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We apply our general results to the study of the problem of existence of different classes of Poisson stable solutions and global attractors for a chemical reaction network and nonautonomous translation-invariant difference equations.

Strongly compact strong trajectory attractors for evolutionary systems and their applications

We show that for any fixed accuracy and time length T, a finite number of T-time length pieces of the complete trajectories on the global attractor are capable of uniformly approximating all trajectories within the accuracy in the natural strong metric after sufficiently large time when the observed dissipative system is asymptotically compact. Moreover, we obtain the strong equicontinuity of all the complete trajectories on the global attractor. These results follow by proving the existence of a strongly compact strong trajectory attractor. The notion of a trajectory attractor was previously constructed for a family of auxiliary systems including the originally considered one without uniqueness. Recently, Cheskidov and the author developed a new framework called evolutionary system, with which a (weak) trajectory attractor can be actually defined for the original system. In this paper, the theory of trajectory attractors is further developed in the natural strong metric for our purpose. We then apply it to both the 2D and the 3D Navier–Stokes equations and a general nonautonomous reaction–diffusion system.

Hyperdissipative Navier–Stokes Equations Driven by Time-Dependent Forces: Invariant Manifolds

We consider the incompressible hyperdissipative Navier–Stokes equations \begin{align}\left \{\begin{array}{l} u_t+\epsilon (-\Delta )^\alpha u+(u\cdot \nabla )u+\nabla p=f,\\ \nabla \cdot u=0, \end{array} \right .\end{align} on a two- or three-dimensional periodic torus, where the power and the forcing function is time-dependent. We intend to reveal how the fractional dissipation and the time-dependent force affect long-time dynamics of weak solutions. More precisely, we prove that with certain conditions on , there exists a finite-dimensional Lipschitz manifold in the -space of divergent-free vector fields with zero mean. The manifold is locally forward invariant and pullback exponentially attracting. Moreover, the compact uniform attractor is contained in the union of all fibers of the manifold. In our result, no large viscosity is assumed. It is also significant that in the three-dimensional case the spectrum of the fractional Laplacian does not have arbitrarily large gaps.

The Rayleigh–Bénard Problem for Compressible Fluid Flows

We consider the physically relevant fully compressible setting of the Rayleigh–Bénard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In addition, global-in-time trajectories are asymptotically compact in suitable topologies and the system possesses a global compact trajectory attractor mathcal {A}. The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure—a stationary statistical solution sitting on mathcal {A}. In addition, the Birkhoff–Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to mathcal {A} a.s. with respect to the invariant measure.

Asymptotically autonomous dynamics for non-autonomous stochastic $ g $-Navier-Stokes equation with additive noise

&lt;p style='text-indent:20px;'&gt;Both sufficient and necessary criteria for the existence of a bi-parametric attractor which attaches with forward compactness is established. Meanwhile, we prove that, under certain conditions, the components of the random attractor of a non-autonomous dynamical system can converge in time to those of the random attractor of the limiting autonomous dynamical system. As an application of the abstract theory, we show that the non-autonomous stochastic &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ g $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Navier-Stokes (g-NS) equation possesses a forward compact random attractor such that it is forward asymptotically autonomous to a random attractor of the autonomous g-NS equation.&lt;/p&gt;

Global attractors for a full von Karman beam transmission problem

A nonlinear transmisson problem for an elastic full von Karman beam is considered here. We prove that the system possesses a compact global attractor.

Attractor of a nonlinear hybrid reaction–diffusion model of neuroendocrine transdifferentiation of human prostate cancer cells with time-lags

&lt;abstract&gt;&lt;p&gt;Prostate cancer is a serious disease that endangers men's health. The genetic mechanism and treatment of prostate cancer have attracted the attention of scientists. In this paper, we focus on the nonlinear mixed reaction diffusion dynamics model of neuroendocrine transdifferentiation of prostate cancer cells with time delays, and reveal the evolutionary mechanism of cancer cells mathematically. By applying operator semigroup theory and the comparison principle of parabolic equation, we study the global existence, uniqueness and boundedness of the positive solution for the model. Additionally, the global invariant set and compact attractor of the positive solution are obtained by Kuratowski's measure of noncompactness. Finally, we use the Pdepe toolbox of MATLAB to carry out numerical calculations and simulations on an example to check the correctness and effectiveness of our main results. Our results show that the delay has no effect on the existence, uniqueness, boundedness and invariant set of the solution, but will affect the attractor.&lt;/p&gt;&lt;/abstract&gt;

Threshold asymptotic dynamics for a spatial age-dependent cell-to-cell transmission model with nonlocal disperse

This research is concerned to study the global threshold dynamics of a hybrid viral infection model, with the assumption that all parameters are space dependent. The considered hybrid model contains three principal equations, the first is an ordinary differential equation that represents the uninfected cells, an age-structured equation modeled by the transport equation for infected-cells, and a diffusive equation that studies the evolution of the virus. The challenging mathematical aspect of this research lies in the fact that the model is partially degenerate and the solution map is not compact. Also, understanding the global threshold dynamics in terms of this partial degeneracy, mostly with age-structured equation and nonlocal diffusion is the ultimate challenging point of this research. The existence and the uniqueness of a global solution are proved. Further, the existence of a global compact attractor is shown. Moreover, the basic reproduction number $ \mathfrak{R}_0 $ is identified for the model with its threshold role: If $ \mathfrak{R}_0&lt;1 $, the infection-free-steady state is globally asymptotically stable; for $ \mathfrak{R}_0&gt;1 $, the solution of the studied model is uniformly persistent and the pathogen persists in a unique positive steady state, which it is shown using the Lyapunov approach. The results are supported by some graphical representations for illustrating the finding.

Global attractor for a degenerate Klein–Gordon–Schrödinger-type system

In this paper, we study the long-time behaviour of solutions of a degenerate Klein–Gordon–Schrödinger-type system which is defined in a bounded domain. First, we proved the existence, uniqueness and continuity of the solutions on the initial data, then the asymptotic compactness of the solutions and finally the existence of a global compact attractor.

Global Attractors for a Class of Weakly Damped Wave Equations with Gradient Type Nonlinearity

In this paper, the main purpose is to study existence of the global attractor for the weakly damped wave equation with gradient type nonlinearity. To this end, we first verify the existence and uniqueness of global weak solution by the Galerkin method and compulsively variational method. Furthermore, we obtained the global strong solution under some mild assumptions on f. Secondly, we utilize the omega-limit compactness to show the semigroup generated by the equation has a compact, connected and invariant attractor.

Attractors for the inhomogeneous incompressible Navier–Stokes flows

This paper is devoted to the evolution of Lions’s weak solutions to the inhomogeneous Navier–Stokes equations. After proving that the kinetic energy is eventually bounded, we obtain a weakly compact global attractor that all Lions’s weak solutions approach as time tends to infinity. Furthermore, the existence of attracting sets in strong topology is established for short trajectories satisfying an additional compactness condition on the density.