Existence Of Pullback Attractors Research Articles

Non-Newtonian incompressible fluids with nonlinear shear tensor and hereditary conditions

We consider a mathematical model with delay for non-Newtonian incompressible fluids in a bounded domain. Existence of global weak solutions is proved under suitable regularity on the initial data and the forces. Conditions for uniqueness are also given, but in general the results are stated in a multi-valued framework. Suitable multi-valued dynamical systems are well-posed, using basically L2(Ω)n×L2(−h,0;L2(Ω)n) or C([−h,0];L2(Ω)n) norms. Then the existence of pullback attractors is ensured acting on several universes, some of them of fixed bounded sets and others of tempered type, depending on parameters related to an integrability condition of the force and the delay term. Finally, relationships between these families of attractors are also provided, improving the characterization of attraction with respect to previous results.

Long-time behavior of nonclassical diffusion equations with memory on time-dependent spaces

This paper aims to study the long-time behavior of nonclassical diffusion equation with memory and disturbance parameters on time-dependent space. By using the contractive process method on the family of time-dependent spaces and operator decomposition technique, the existence of pullback attractors is first proved. Then the upper semi-continuity of pullback attractors with respect to perturbation parameter ν in M t is obtained. It’s remarkable that the nonlinearity f satisfies the polynomial growth of arbitrary order.

Pullback Attractors of the Bingham Model

Based on the theory of trajectory pullback attractors, we study the qualitative behavior of weak solutions for the Bingham model with periodic conditions in the space variables. For the model under consideration, a family of trajectory spaces is introduced and the existence of pullback attractors is proved.

Robustness of 3D Navier–Stokes System with Increasing Damping

The principal objective of the paper is the study of the three-dimensional Navier–Stokes system with non-autonomous perturbation force term and increasing damping term, which often appears in the fluid system within saturated porous media and other complex media. With some suitable assumptions on the system parameters and external force term, based on the known result on global well-posedness, the existence of pullback attractors is educed, and the system robustness is shown via the upper semicontinuity of system attractors as the perturbation parameter approaches a certain value.

A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient

<p style='text-indent:20px;'>A mathematical model describing the growth of gut microbiome inside and on the wall of the gut is developed based on the chemostat model with wall growth. Both the concentration and flow rate of the nutrient input are time-dependent, which results in a system of non-autonomous differential equations. First the stability of each meaningful equilibrium is studied for the autonomous counterpart. Then the existence of pullback attractors and its detailed structures for the nonautonomous system are investigated using theory and techniques of nonautonomous dynamical systems. In particular, sufficient conditions under which the microbiome vanishes or persists are constructed. Numerical simulations are provided to illustrate the theoretical results.</p>

Pullback Attractors for Nonautonomous Degenerate Kirchhoff Equations with Strong Damping

In this paper, we obtain the existence of pullback attractors for nonautonomous Kirchhoff equations with strong damping, which covers the case of possible generation of the stiffness coefficient. For this purpose, a necessary method via “the measure of noncompactness” is established.

Sigmoidal Approximations of a Nonautonomous Neural Network with Infinite Delay and Heaviside Function

In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attractors in both cases, and the convergence of the attractors of the sigmoidal models to those of the Heaviside inclusion.

Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays

<p style='text-indent:20px;'>In this paper, we investigate a class of stochastic recurrent neural networks with discrete and distributed delays for both biological and mathematical interests. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions so that the uniqueness of the Cauchy problem fails to be true. Moreover, the existence of pullback attractors with or without periodicity is presented for the multi-valued noncompact random dynamical system. In particular, a new method for checking the asymptotical compactness of solutions to the class of nonautonomous stochastic lattice systems with infinite delay is used.</p>

Pullback Attractors for a Nonautonomous Retarded Degenerate Parabolic Equation

This paper is devoted to a nonautonomous retarded degenerate parabolic equation. We first show the existence and uniqueness of a weak solution for the equation by using the standard Galerkin method. Then we establish the existence of pullback attractors for the equation by proving the existence of compact pullback absorbing sets and the pullback asymptotic compactness.

Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions

In this paper we study the asymptotic behavior of solutions for a class of nonautonomous reaction-diffusion equations with dynamic boundary conditions possessing finite delay. Under the polynomial conditions of reaction term, suitable conditions of delay terms and a minimal conditions of time-dependent force functions, we first prove the existence and uniqueness of solutions by using the Galerkin method. Then, we ensure the existence of pullback attractors for the associated process to the problem by proving some uniform estimates and asymptotic compactness properties (via an energy method). With an additional condition of time-dependent force functions, we prove that the boundedness of pullback attractors in smoother spaces.

Strong Pullback Attractors of the Nonautonomous Suspension Bridge Equation with Variable Delay

In this paper, we consider the suspension bridge equation with variable delay. The long-time dynamics of the solutions for the suspension equations without delay effects have been investigated by many authors. But there are few works on suspension equations with delay. Moreover there are not many studies on attractors for other systems with delay. Thus, we study the existence of pullback attractor for the suspension equation with variable delay by using the theory of attractors for multivalued dynamical systems.

Pullback attractors for non-autonomous reaction\u2013diffusion equation with infinite delays in C_{gamma,L^{r}(Omega)} or C_{gamma,W^{1,r}(Omega)}

In this paper, the well-posedness for the non-autonomous reaction–diffusion equation with infinite delays on a bounded domain is established. The existence of pullback attractors for the process in C_{gamma,L^{r}(Omega)} and C_{gamma,W^{1,r}(Omega)} is proved, respectively. The noncompact Kuratowski measure is applied to check the asymptotic compactness.

Dynamics of non-autonomous stochastic dissipative Hamiltonian amplitude wave equations with rapidly oscillating force

ABSTRACTThis paper considers the limiting behavior of pullback attractors for the non-autonomous stochastic dissipative Hamiltonian amplitude wave equation with rapidly oscillating force defined on a real line. The presence of the complex valued term brings some difficulties even to prove the induced system is pullback absorbing. Thus, by choosing a new number , some new uniform asymptotic estimates are used to prove the existence of pullback attractors in , as well as to show the upper semicontinuity of the family of oscillating attractors when tends to the average as the density of noise approaches to zero.

Pullback attractors of 2D incompressible Navier‐Stokes‐Voight equations with delay

In this paper, the 2D Navier‐Stokes‐Voight equations with 3 delays in is considered. By using the Faedo‐Galerkin method, Lions‐Aubin lemma, and Arzelà‐Ascoli theorem, we establish the global well‐posedness of solutions and the existence of pullback attractors in H1.

Pullback Asymptotic Behavior of Solutions for a 2D Non-autonomous Non-Newtonian Fluid

This paper studies the pullback asymptotic behavior of solutions for the non-autonomous incompressible non-Newtonian fluid in 2D bounded domains. Firstly, with a little high regularity of the force, the semigroup method and \(\epsilon \)-regularity method are used to establish the existence of compact pullback absorbing sets. Then, with a minimal regularity of the force, by verifying the flattening property also known as the “Condition (C)”, the author proves the existence of pullback attractors for the universe of fixed bounded sets and for the another universe given by a tempered condition. Furthermore, the regularity of pullback attractors is given.

Pullback D $\mathcal{D}$ -attractors for three-dimensional Navier-Stokes equations with nonlinear damping

We investigate the asymptotic behavior of solutions of the non-autonomous Navier-Stokes equation with nonlinear damping in three-dimensional bounded domain. When $3< \beta\leq5$ , the existence of pullback attractors is proved in V and $\mathbf{H}^{2}(\Omega)$ , respectively.

Pullback attractors of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows

This paper is concerned with the long-time behaviour of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows introduced in Lin and Liu (Commun Pure Appl Math, 48:501–537, 1995) with a non-autonomous forcing bulk term and order parameter field boundary conditions. In this paper, we prove the existence of pullback attractors and estimate the upper bound of its fractal dimension under some suitable assumptions.

Dynamics of stochastic Boissonade system on the time-varying domain

The current paper is devoted to studying the stochastic Boissonade system defined on time-varying domains. The existence and uniqueness of strong and weak solutions for the stochastic Boissonade system are established. Moreover, the existence of pullback attractor for the ‘partial-random’ system generated by the weak solution is also presented.

Pullback attractors of 2D Navier-Stokes-Voigt equations with delay on a non-smooth domain

Under suitable hypotheses on the continuous delay, distributed delay, and the initial data in this paper, the large-time behavior for the 2D Navier-Stokes-Voigt equations with continuous delay and distributed delay on the Lipschitz domain is studied. The existence of pullback attractors in the non-smooth domain was obtained via verifying some pullback dissipation and asymptotical compactness for the continuous process.

Approximate Kelvin-Voigt Fluid Driven by an External Force Depending on Velocity with Distributed Delay

We consider the approximate 3D Kelvin-Voigt fluid driven by an external force depending on velocity with distributed delay. We investigate the long time behavior of solutions to Navier-Stokes-Voigt equation with a distributed delay external force depending on the velocity of fluid on a bounded domain. By a prior estimate and a contractive function, we give a sufficient condition for the existence of pullback attractor of NSV equation.