Upper Semicontinuity Of Attractors Research Articles

Synchronization of stochastic lattice equations and upper semicontinuity of attractors

We consider a system of two coupled stochastic lattice equations driven by additive white noise processes, where the strength of the coupling is given by a parameter We show that these equations generate a random dynamical system which has a random pullback attractor. This attractor naturally depends on the parameter κ. When the intensity of the coupling becomes large, we observe that the components of the given system synchronize. To describe this phenomenon, we prove the upper semicontinuity of the family of attractors with respect to the attractor of a specific limiting system.

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p-1 (p geq 2). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

Random dynamics for non-autonomous stochastic evolution equations without uniqueness on unbounded narrow domains

This paper deals with the limiting behavior of non-autonomous stochastic reaction-diffusion equations without uniqueness on unbounded narrow domains. We prove the existence and upper semicontinuity of random attractors for the equations on a family of unbounded (n + 1)-dimensional narrow domains, which collapses onto an n-dimensional domain. Since the solutions are non-uniqueness, which leads to a multivalued random dynamical system with the solution operators of the equation, we will prove the existence and upper semicontinuity of attractors by multivalued random dynamical system theory.

Limiting dynamics for stochastic reaction–diffusion equations on the Sobolev space with thin domains

This paper is devoted to bi-spatial random attractors of the stochastic reaction–diffusion equation when the terminate space is the Sobolev space on a thin domain, where the nonlinearity can be decomposed into two functions with (p,q)-growth exponents. By means of a computation method of induction, it is shown that the difference of solutions near the initial time is integrable of kp−2k+2 order. This higher-order integrability shows continuity of the solution operator from the square Lebesgue space to (kp−2k+2)-order Lebesgue spaces. In particular, the (2p−2)-order integrability shows continuity of the solution operator from the square Lebesgue space to the Sobolev space, which further shows existence of a random attractor in the Sobolev space when the initial space is the square Lebesgue space. Moreover, the higher-order integrability can be uniform with respect to all thin domains, which provides uniformly asymptotic compactness of the random dynamical systems. As a conclusion, the upper semi-continuity of attractors under the Sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain.

Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise

This paper is concerned with the regular random dynamics for the reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where the usual Winner process is replaced by a general stochastic process satisfied the basic convergence. A bi-spatial attractor is obtained when the non-initial space is $p$-times Lebesgue space or Sobolev space. The measurability of the solution operator is proved, which leads to the measurability of the attractor in both state spaces. Finally, the upper semi-continuity of attractors under the $p$-norm is established when the narrow domain degenerates onto a lower dimensional set. Both methods of symbolical truncation and spectral decomposition provide all required uniform estimates in both Lebesgue and Sobolev spaces.

Pullback Dynamics of Non-autonomous Timoshenko Systems

This paper is concerned with the Timoshenko system, a recognized model for vibrations of thin prismatic beams. The corresponding autonomous system has been widely studied. However, there are only a few works dedicated to its non-autonomous counterpart. Here, we investigate the long-time dynamics of Timoshenko systems involving a nonlinear foundation and subjected to perturbations of time-dependent external forces. The main result establishes the existence of a pullback exponential attractor, which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. The upper-semicontinuity of attractors, as the non-autonomous forces tend to zero, is also studied.

Existence and Upper Semicontinuity of Attractors for Nonautonomous Stochastic Sine-Gordon Lattice Systems with Random Coupled Coefficients

We study nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise. We first consider the existence of random attractors in a weighted space for this system and then establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Upper Semicontinuity of Attractors for a Non-Newtonian Fluid under Small Random Perturbations

This paper investigates the limiting behavior of attractors for a two-dimensional incompressible non-Newtonian fluid under small random perturbations. Under certain conditions, the upper semicontinuity of the attractors for diminishing perturbations is shown.

Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity

In this study we consider the von Karman evolution equations accounting for rotational inertial forces along with a nonlinear localized damping. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact attractor Aα,R. The main result of the paper is upper semicontinuity of attractors with respect to rotational inertia terms. The main obstacles are: criticality of the nonlinear term of the equation combined with geometrically constrained dissipation. The flux multiplier used for propagation of the damping does not cooperate with the nonlinear term which is of critical exponent. A new method based on uniform compactness is developed in order to handle such cases. The method allows to establish uniform smoothness of the elements from the attractor which is the key ingredient in proving that the attractor is upper semicontinuous with respect to the rotational inertia α>0 and conclude that it “converges” to the attractor AR obtained (in the case α=0) in Geredeli et al. (2013) [23] when α→0.

DYNAMICAL BEHAVIOR OF THE ALMOST-PERIODIC DISCRETE FITZHUGH–NAGUMO SYSTEMS

In this paper, we study the dynamical behavior of nonautonomous, almost-periodic discrete FitzHugh–Nagumo system defined on infinite lattices. We prove that the nonautonomous infinite-dimensional system has a uniform attractor which attracts all solutions uniformly with respect to the translations of external terms. We also establish the upper semicontinuity of uniform attractors when the infinite-dimensional system is approached by a family of finite-dimensional systems. This paper is based on a uniform tail method, which shows that, for large time, the tails of solutions are uniformly small with respect to bounded initial data as well as the translations of external terms. The uniform tail estimates play a crucial role for proving the uniform asymptotic compactness of the system and the upper semicontinuity of attractors.

Upper semi-continuity of attractors for multivalued semi-flow under random perturbation

In this paper the upper semi-continuity of global attractors for multivalued semi-flows under random perturbation was studied. First, the existence of random attractors for multivalued random semi-flows was considered, then it was proved that the global attractors for multivalue semi-flows are the upper semi-continuity under random perturbation. This result can be used in the numerical approximation of multivalued semi-flows and non-autonomous perturbation of multivalued semi-flows.

Upper semi continuity of attractors of delay differential equations in the delay

It is shown that if a retarded delay differential equation has a global attractor in the space C ([—τ0, ], ℝd) for a given nonzero constant delay τ0, then the equation has an attractor Aτ in the space C ([—τ, 0], ℝd) for nearby constant delays τ. Moreover the attractors Aτ converge upper semi continuously to in C ([—τ0, 0], ℝd) in the sense that they are identified through corresponding segments of entire trajectories in ℝd with nonempty compact subsets of C ([—τ0, 0], ℝd) which converge upper semi continuously to in C ([—τ0, 0], ℝd).

Set value mapping and its application

The dynamics of set value mapping is considered. For the upper semi-continuous set value maps, the existence of attractors under some conditions and the upper semi-continuity of attractors under the perturbation are proved. Its application in numerical simulation of differential equation is also considered. The upper semi-continuity of attractors in set value maps under the perturbation is used to show the reasonable of subdivision algorithm and interval arithmetic in numerical simulation of differential equation.

Upper semicontinuity of attractors for a semilinear reaction and diffusion system of equations, in sup norm

Upper semicontinuity of attractors for a semilinear reaction and diffusion system of equations, in sup norm

Perturbation of the diffusion and upper semicontinuity of attractors

We obtain uniform bounds for attractors of parabolic problems with nonlinear boundary conditions with respect to perturbations in the diffusion coefficients. These bounds are the main tool to obtain continuity properties of the attractors relatively to perturbations of the diffusion coefficient. We study two examples where the diffusion is perturbed: one related to homogenization theory and the other taken from composite materials where the diffusion goes to infinity in some subdomain.

Upper Semicontinuity of Attractors and Synchronization

In this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize. We apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems. The technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem, explicitly exhibited, in the diagonal.

On the perturbation of the kernel for delay systems with continuous kernels

In this paper we investigate upper semicontinuity of attractors for delay equations where the delay depends on a parameter. We are especially interested in delay equations with continuous memory functions. We interpret the results of Hines[l2] in this context, giving conditions on the memory functions under which we have upper semicontinuity. We then present some examples from the literature where we use these results to prove that attractors are upper semicontinuous.