Exponential Attractors Research Articles

The Family of Exponential Attractors and Inertial Manifolds for a Generalized Nonlinear Kirchhoff Equations

In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and then the existence of its exponential attractor is obtained. By extending the space E0 to Ek, a family of the exponential attractors of the initial-boundary value problem is obtained. In the second part, we consider the long-time behavior for a system of generalized Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.

On the modified of the one-dimensional Cahn-Hilliard equation with a source term

<abstract><p>We consider the modified Cahn-Hilliard equation that govern the relative concentration $ \phi $ of one component of a binary system. This equation is characterized by the presence of the additional inertial term $ \tau_{D}\frac{d^2\phi}{dt^2} $ which stands for the relaxation of the diffusion flux. This equation is associated with Dirichlet boundary conditions. We study the existence, uniqueness and regularity of solutions in one space dimension. We also prove the existence of the global attractor and exponential attractors.</p></abstract>

Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay

<p style='text-indent:20px;'>In this paper, we study the long term dynamical behavior of a two-dimensional nonlocal diffusion lattice system with delay. First some sufficient conditions for the construction of an exponential attractor are presented for infinite dimensional autonomous dynamical systems with delay. Then, the existence of exponential attractors for the two-dimensional nonlocal diffusion delay lattice system is established by using the new method of tail-estimates of solutions and overcoming the difficulties caused by the nonlocal diffusion operator and the multi-dimensionality.</p>

Strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation

<p style='text-indent:20px;'>In this paper, we investigate the global well-posedness and the existence of strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation in <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset{\mathbb R}^N $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}+\Delta ^2u+\Delta ^2u_t+\Delta \phi (\Delta u) = g(x), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with the hinged boundary condition. We show that (i) when the nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> is quasi-monotone and is of at most the critical growth: <inline-formula><tex-math id="M3">\begin{document}$ 1\leq p\leq p^{*}: = \frac{N+2}{(N-2)^{+}} \ (N\geq 2) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ g = 0 $\end{document}</tex-math></inline-formula>, the model has in phase space <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal H} = V_3\times L^2 $\end{document}</tex-math></inline-formula> a trivial global and exponential attractor, respectively. (ii) In particular when <inline-formula><tex-math id="M6">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>, without any polynomial growth restriction for <inline-formula><tex-math id="M7">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>, the model has a strong global and a strong exponential attractor, respectively. These results deepen and extend the related researches on this topic in recent literature [<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b22">22</xref>]. The method developed here allows us to establish the existence of the strong global and exponential attractor for this nonlinear model.</p>

Long-time dynamical behavior for a piezoelectric system with magnetic effect and nonlinear dampings

<abstract><p>This paper is concerned with the long-time dynamical behavior of a piezoelectric system with magnetic effect, which has nonlinear damping terms and external forces with a parameter. At first, we use the nonlinear semigroup theory to prove the well-posedness of solutions. Then, we investigate the properties of global attractors and the existence of exponential attractors. Finally, the upper semicontinuity of global attractors has been investigated.</p></abstract>

Attractors for the Navier-Stokes-Cahn-Hilliard system

<p style='text-indent:20px;'>We investigate the longtime behavior of the solutions to the Navier-Stokes-Cahn-Hilliard system (also known as Model H) with singular (e.g. Flory-Huggins) potential and non-constant viscosity. We prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space. Next, we establish the existence of the global attractor and of exponential attractors, and their regularity properties.</p>

Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems

<p style='text-indent:20px;'>In this paper we study long-time behavior of first-order non-autono-mous lattice dynamical systems in square summable space of double-sided sequences using the cooperation between the discretized diffusion operator and the discretized reaction term. We obtain existence of a pullback global attractor and construct pullback exponential attractor applying the introduced notion of quasi-stability of the corresponding evolution process.</p>

Long time dynamics and upper semi-continuity of attractors for piezoelectric beams with nonlinear boundary feedback

A system of boundary-controlled piezoelectric beam equations, accounting for the interactions between mechanical vibrations and the fully-dynamic electromagnetic fields, is considered. Even though electrostatic and quasi-static electromagnetic field approximations of Maxwell’s equations are sufficient for most models of piezoelectric systems, where the magnetic permeability is completely discarded, the PDE model considered here retains the pronounced wave behavior of electromagnetic fields to accurately describe the dynamics for the most piezoelectric acoustic devices. It is also crucial to investigate whether the closed-loop dynamics of the fully-dynamic piezoelectric beam equations, with nonlinear state feedback and nonlinear external sources, is close to the one described by the electrostatic/quasi-static equations, when the magnetic permeability μ is small. Therefore, the asymptotic behavior is analyzed for the fully-dynamic model at first. The existence of global attractors with finite fractal dimension and the existence of exponential attractors are proved. Finally, the upper-semicontinuity of attractors with respect to magnetic permeability to the ones of the electrostatic/quasi-static beam equations is shown.

Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise

<p style='text-indent:20px;'>In this paper, we mainly consider the existence of random attractor and random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. First step, the well-posedness and the existence of a random attractor for the cocycle associated with the considered system is established. Second step, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero. Third step, we prove the regularity of random attractor in a higher regular space by the "iteration" method. Finally, we give the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor.</p>

On a generalized diffusion problem: A complex network approach

<p style='text-indent:20px;'>In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution <i>via</i> operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors.</p>

Exponential attractor for a class of non local evolution equations

Exponential attractor for a class of non local evolution equations

A Family of Exponential Attractors and Inertial Manifolds for a Class of Higher Order Kirchhoff Equations

In this paper, the global dynamics of a class of higher order nonlinear Kirchhoff equations under n-dimensional conditions is studied. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup associated with the initial boundary value problem are proved, and the existence of a family of exponential attractors is obtained. Then, by constructing the corresponding graph norm, the condition of a spectral interval is established when N is sufficiently large. Finally, the existence of the family of inertial manifolds is obtained.

On the viscous Cahn–Hilliard–Oono system with chemotaxis and singular potential

We analyze a diffuse interface model that couples a viscous Cahn–Hilliard equation for the phase function φ with a diffusion‐reaction equation for the nutrient concentration σ. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport, and nonlocal interaction of Oono's type. When the spatial dimension is three, we prove the existence and uniqueness of a global weak solution to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solution when t > 0. In particular, with the aid of the viscous term ϵ∂tφ, we prove the so‐called instantaneous separation property of the phase variable φ such that it stays away from the pure states ±1 as long as t > 0. Next, we study longtime behavior of the system, by proving the existence of the global attractor in a suitable phase space and characterizing the ω‐limit set. Moreover, we show the existence of an exponential attractor, which implies that the global attractor is of finite fractal dimension.

The effect of self-limiting on the prevention and control of the diffuse COVID-19 epidemic with delayed and temporal-spatial heterogeneous

BackgroundThe global spread of the novel coronavirus pneumonia is still continuing, and a new round of more serious outbreaks has even begun in some countries. In this context, this paper studies the dynamics of a type of delayed reaction-diffusion novel coronavirus pneumonia model with relapse and self-limiting treatment in a temporal-spatial heterogeneous environment.MethodsFirst, focus on the self-limiting characteristics of COVID-19, incorporate the relapse and self-limiting treatment factors into the diffusion model, and study the influence of self-limiting treatment on the diffusion of the epidemic. Second, because the traditional Lyapunov stability method is difficult to determine the spread of the epidemic with relapse and self-limiting treatment, we introduce a completely different method, relying on the existence conditions of the exponential attractor of our newly established in the infinite-dimensional dynamic system to determine the diffusion of novel coronavirus pneumonia. Third, relapse and self-limiting treatment have led to a change in the structure of the delayed diffusion COVID-19 model, and the traditional basic reproduction number R_0 no longer has threshold characteristics. With the help of the Krein-Rutman theorem and the eigenvalue method, we studied the threshold characteristics of the principal eigenvalue and found that it can be used as a new threshold to describe the diffusion of the epidemic.ResultsOur results prove that the principal eigenvalue uplambda ^{*} of the delayed reaction-diffusion COVID-19 system with relapse and self-limiting treatment can replace the basic reproduction number R_0 to describe the threshold effect of disease transmission. Combine with the latest official data and the prevention and control strategies, some numerical simulations on the stability and global exponential attractiveness of the diffusion of the COVID-19 epidemic in China and the USA are given.ConclusionsThrough the comparison of numerical simulations, we find that self-limiting treatment can significantly promote the prevention and control of the epidemic. And if the free activities of asymptomatic infected persons are not restricted, it will seriously hinder the progress of epidemic prevention and control.

Asymptotic analysis and upper semicontinuity to a system of coupled nonlinear wave equations

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.

Long-time dynamics of the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions

We consider a Cahn–Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf et al. (2021) and will thus be called the KLLM model. In the aforementioned paper, it was shown that solutions of the KLLM model converge to solutions of the GMS model proposed by Goldstein et al. (2011) as the kinetic rate tends to infinity. We first collect the weak well-posedness results for both models and we establish some further essential properties of the weak solutions. Afterwards, we investigate the long-time behavior of the KLLM model. We first prove the existence of a global attractor as well as convergence to a single stationary point. Then, we show that the global attractor of the GMS model is stable with respect to perturbations of the kinetic rate. Eventually, we construct exponential attractors for both models, and we show that the exponential attractor associated with the GMS model is robust against kinetic rate perturbations.

Global and exponential attractors for mixtures of solids with Fourier’s law

This paper presents a study of the long-time dynamics of the dynamical system generated by a nonlinear system modeling mixture of solids with nonlinear damping and Fourier’s law. By using the recent quasi-stability theory, we prove the existence of a smooth finite dimensional global attractor, which is characterized as an unstable manifold of the set of stationary solutions. The quasi-stability of the system is achieved through an estimated stabilizability. Moreover, the existence of a generalized exponential attractor is shown.

Exponential Attractor for the Viscoelastic Wave Model with Time-Dependent Memory Kernels

The paper is concerned with the exponential attractors for the viscoelastic wave model in $$\varOmega \subset \mathbb R^3$$ : $$\begin{aligned} u_{tt}-h_t(0)\varDelta u-\int _0^\infty \partial _sh_t(s)\varDelta u(t-s)\mathrm ds+f(u)=g, \end{aligned}$$ with time-dependent memory kernel $$h_t(\cdot )$$ which is used to model aging phenomena of the material. Conti et al. (Am J Math 140(2):349–389, 2018a; Am J Math 140(6):1687–1729, 2018b) recently provided the correct mathematical setting for the model and a well-posedness result within the novel theory of dynamical systems acting on time-dependent spaces, recently established by Conti et al. (J Differ Equ 255:1254–1277, 2013), and proved the existence and the regularity of the time-dependent global attractor. In this work, we further study the existence of the time-dependent exponential attractors as well as their regularity. We establish an abstract existence criterion via quasi-stability method introduced originally by Chueshov and Lasiecka (J Dyn Differ Equ 16:469–512, 2004), and on the basis of the theory and technique developed in Conti et al. (2018a, b) we further provide a new method to overcome the difficulty of the lack of further regularity to show the existence of the time-dependent exponential attractor. And these techniques can be used to tackle other hyperbolic models.

Reaction-diffusion systems with supercritical nonlinearities revisited

We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We examine the global existence and uniqueness of weak and strong solutions, various types of smoothing properties, asymptotic compactness and the existence of global and exponential attractors.

Long-time dynamics of a class of nonlocal extensible beams with time delay

This work is devoted to the following nonlocal extensible beam equation with time delay: u t t + Δ 2 u − M ( ∫ Ω | ∇ u | 2 d x ) Δ u + a 0 u t ( x , t ) + a 1 u t ( x , t − τ ) + φ ( u ) = f , on a bounded smooth domain . The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.