Existence Of Exponential Attractor Research Articles

Exponential attractor for Kirchhoff model with time delay and thermal effect

The Kirchhoff model is derived from the vibration problem of stretchable strings. In this paper, we focus on the long-time dynamics of the Kirchhoff model with time delay and thermal effect. We first proved the existence and uniqueness of the solution through the Faedo–Galerkin method and prior estimates. After that, it is further proved that the dynamical system has a bounded absorption set, and it is also verified that the system is quasi-stable on the bounded absorbing set. Finally, the existence of exponential attractor in dynamical system is proved. This innovatively considers the long-time dynamical behavior of Kirchhoff model under the simultaneous action of variable coefficient, thermal effect and time delay comprehensively. It also promotes the relevant conclusions of the Kirchhoff model. The findings of this paper provide a theoretical basis for subsequent application and research.

Exponential attractors for modified Swift-Hohenberg equation in $\mathbb R^{N}$

A Cauchy problem for a modification of the Swift-Hohenberg equation in $\mathbb R^{N}$ with a mildly integrable potential is considered. Existence of exponential attractors containing a finite-dimensional global attractor in $H^2(\mathbb R^{N})$ is shown under the dissipative mechanism of fourth order parabolic equations in unbounded domains.

Dynamics of locally damped Timoshenko systems

In this paper, we study the long-time dynamics of a Timoshenko system modeling vibrations of beams with non-linear localized damping mechanisms acting on both displacement and angular rotation and subjected to non-linear source terms. Using recent quasi-stability methods, we prove the existence of smooth finite dimensional global attractor, which is characterized as unstable manifold of the set of stationary solutions. Moreover, the existence of exponential attractors is shown. These aspects were not previously considered for the Timoshenko system with localized damping.

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>

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.

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.

A Family of the Exponential Attractors and the Inertial Manifolds for a Class of Generalized Kirchhoff Equations

In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation, then according to Hadamard’s graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.

Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

<abstract><p>In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup $ \{S(t)\}_{t\geq 0} $ corresponding to this equation satisfies the global exponentially $ \kappa- $dissipative. And then we estimate the upper bound of fractal dimension for the global attractors $ \mathscr{A} $ for this equation and $ \mathscr{A}\subset H^1_0(\Omega)\cap H^2(\Omega) $. Finally, we confirm the existence of exponential attractors $ \mathscr{M} $ by validated differentiability of the semigroup $ \{S(t)\}_{t\geq 0} $. It is worth mentioning that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.</p></abstract>

Existence and Upper-Semicontinuity of Global Attractors for Binary Mixtures Solids with Fractional Damping

This paper is devoted to study the asymptotic behavior of a binary mixture problem of solids with fractional damping and sources terms. We prove the existence of global attractors with finite fractal dimension and the existence of exponential attractors. Moreover, we prove the upper-semicontinuity of global attractors as the fractional exponent tend to zero.

Attractors for the nonclassical diffusion equations of Kirchhoff type with critical nonlinearity on unbounded domain

ABSTRACTIn this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations of Kirchhoff type First, we prove the well-posedness of solution for the nonclassical diffusion equations of Kirchhoff type with critical nonlinearity on , then the existence of global attractor is established in the natural energy space . Finally, we obtain the existence of exponential attractors in , as a product, the global attractor has finite fractal dimension.

The global attractors and exponential attractors for a class of nonlinear damping Kirchhoff equation

This paper consider the long time behavior of a class of nonlinear damped Kirchhoff equation    2 tt 1 t t u u u u u f x          . Study the attractor problem with initial boundary value conditions, then using priori estimate and the Galerkin method prove existence and uniqueness of solution, we obtain to the existence of the global attractors. The squeezing property of the nonlinear semi-group associated with this equation and the existence of exponential attractors are also proved.

Long-time dynamics for a nonlinear Timoshenko system with delay

This paper is concerned with a nonlinear Timoshenko system with a time delay term in the internal feedback together with initial data and Dirichet boundary conditions. Under some suitable assumptions on the weights of feedback, we obtain the existence of a global attractor with finite fractal dimension for the case of equal speed wave propagation. Furthermore, the existence of exponential attractors is also derived.

Exponential Attractor for the Boussinesq Equation with Strong Damping and Clamped Boundary Condition

The paper studies the existence of exponential attractor for the Boussinesq equation with strong damping and clamped boundary conditionutt-Δu+Δ2u-Δut-Δg(u)=f(x). The main result is concerned with nonlinearitiesg(u)with supercritical growth. In that case, we construct a bounded absorbing set with further regularity and obtain quasi-stability estimates. Then the exponential attractor is established in natural energy spaceV2×H.

Long-time dynamics of an extensible plate equation with thermal memory

This work is concerned with long-time dynamics of solutions of extensible plate equations with thermal memory. The problem corresponds to a model of thermoelasticity based on a theory of non-Fourier heat flux. By considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system and guarantees the existence of a finite-dimensional global attractor. In addition, the existence of exponential attractors is also considered.

Uniform Exponential Attractors for Non-Autonomous Strongly Damped Wave Equations

In this paper, we study the existence of exponential attractors for strongly damped wave equations with a time-dependent driving force. To this end, the uniform H?lder continuity is established to the variation of the process in the phase apace. In a certain parameter region, the exponential attractor is a uniformly exponentially attracting time-dependent set in the phase apace, and is finite-dimensional no matter how complex the dependence of the external forces on time is. On this basis, we also obtain the existence of the infinite-dimensional uniform exponential attractor for the system.

Well-posedness and long time behavior of an Allen-Cahn type equation

The aim of this article is to study the existence and uniqueness of solutions for an equation of Allen-Cahn type and to prove the existence of the finite-dimensional global attractor as well as the existence of exponential attractors.

Exponential Attractors in Strong Topology Space for a Nonclassical Diffusion Equation

In this article,we study the existence of exponential attractors in the strong topology space H~2(Ω) n H_0~1(Ω) for the nonclassical Diffusion Equation u_t — △_ut- △_u = f(u) + g(x) with critical exponent.In particular,after proving the existence of exponential attractors,we know that the global attractor in the strong topology space in[7,12,14]with finiteness of fractal dimension.

Uniform global attractors for non-isothermal viscous and non-viscous Cahn–Hilliard equations with dynamic boundary conditions

We consider a model of non-isothermal phase separation taking place in a confined container. The order parameter ϕ is governed by a viscous or non-viscous Cahn–Hilliard type equation which is coupled with a heat equation for the temperature θ . The former is subject to a non-linear dynamic boundary condition recently proposed by physicists to account for interactions with the walls, while the latter is endowed with a standard (Dirichlet, Neumann or Robin) boundary condition. We indicate by α the viscosity coefficient, by ε a (small) relaxation parameter multiplying ∂ t θ in the heat equation and by δ a small latent heat coefficient (satisfying δ ≤ λ α , λ > 0 ) multiplying Δ θ in the Cahn–Hilliard equation and ∂ t ϕ in the heat equation. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. We first prove that the model generates a strongly continuous semigroup on a suitable phase space Y K α (depending on the choice of the boundary conditions) which possesses the global attractor A ε , δ , α . Our main results allow us to show that a proper lifting A 0 , 0 , α , α > 0 , of the global attractor of the well-known viscous Cahn–Hilliard equation (that is, the system corresponding to ( ε , δ ) = ( 0 , 0 ) ) is upper semicontinuous at ( 0 , 0 ) with respect to the family { A ε , δ , α } ε , δ , α > 0 . We also establish that the global attractor A 0 , 0 , 0 of the non-viscous Cahn–Hilliard equation (corresponding to ( ε , α ) = ( 0 , 0 ) ) is upper semicontinuous at ( 0 , 0 ) with respect to the same family of global attractors. Finally, the existence of exponential attractors M ε , δ , α is also obtained in the cases ε ≠ 0 , δ ≠ 0 , α ≠ 0 , ( 0 , δ , α ) , δ ≠ 0 , α ≠ 0 and ( ε , δ , α ) = ( 0 , 0 , α ) , α ≥ 0 , respectively. This allows us to infer that, for each ( ε , δ , α ) ∈ [ 0 , ε 0 ] × [ 0 , δ 0 ] × [ 0 , α 0 ] , A ε , δ , α has finite fractal dimension and this dimension is bounded, uniformly with respect to ε , δ and α .

Convergence to steady state and attractors for doubly nonlinear equations

This paper is concerned with the asymptotic behaviour of a class of doubly nonlinear parabolic-type equations. For degenerate equations, we prove thanks to the Lojasiewicz inequality that the solutions to some nonautonomous equations converge to a steady state. In the nondegenerate case, the existence of exponential attractors is shown by using the ℓ-trajectories method.

Exponential attractor for the generalized symmetric regularized long wave equation with damping term

The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semigroup associated with this equation and the existence of exponential attractor are proved. The upper bounds of its fractal dimension are also estimated.