Existence Of Global Attractor Research Articles

Stability of Solutions to a Caginalp Phase-Field Type Equations

This paper is concerned with the study of the asymptotic behavior of a generalization of the Caginalp phase-field model subject to homogeneous Neumann boundary conditions and regular potentials involving two temperatures. This work follows on from a paper in which the well-posedness of the problem, the dissipativity of the system, and the existence of global and exponential attractors were demonstrated. In addition, a study on the semi-infinite cylinder was also carried out. Indeed, if it is true that the existence of a global attractor makes it possible to predict the asymptotic behavior of solutions on a bounded domain, it does not say that these solutions converge. After having shown the existence of the global attractor, it is therefore important to look at the convergence of the solutions over time. There are several methods for determining the asymptotic behavior of the solutions of a differential system. We can mention the one that consists of transforming the given differential equations into integral equations and then applying the classical Picard successive approximation procedure to them. This work is devoted to the study of the convergence of solutions to steady states, adapting a well-known result concerning Lojasiewicz-Simon’s inequality.

Global attractor and l^p solutions to initial value problems of discrete nonlinear Schrodinger equations complex potential

In this article, we investigate the global well-posedness of initial value problems of the time-dependent discrete nonlinear Schrodinger equation with a complex potential and sufficiently general nonlinearity on a multidimensional lattice in weighted \( l^p\) spaces for \( 1< p <\infty\). Thanks to our improved estimates we are able to prove the existence of global attractor for \( l^p\) solutions to the initial value problem.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/12/abstr.html

Global asymptotic stability for Gurtin-MacCamy’s population dynamics model

In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker’s type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semiflow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter.

Nonlinear dynamical analysis for globally modified incompressible non-Newtonian fluids

We present the global modification of the Ladyzhenskaya equations, for incompressible non-Newtonian fluids. This modification is through a cut-off function that multiplies the convective term of the equation and an additional artificial smoothing dissipation term as part of the viscous term of the equation. The goal of this work is the comparative analysis between the modified system and the non-modified system. Therefore, we show the existence and regularity of weak solutions, the existence of global attractors, the estimation of the fractal dimension of the global attractors, and finally, the relationship of the autonomous dynamics between the modified system and the non-modified system.

Existence of global attractors and convergence of solutions for the Cahn-Hilliard equation on manifolds with conical singularities

We consider the Cahn-Hilliard equation on manifolds with conical singularities and prove existence of global attractors in higher order Mellin-Sobolev spaces with asymptotics. We also show convergence of solutions in the same spaces to an equilibrium point and provide asymptotic behavior of the equilibrium near the conical tips in terms of the local geometry.

Long-time behavior for impulsive generalized semiflows

We propose new criteria, which are more general than those previously known in the literature, to guarantee the existence of global attractors for problems with state-dependent impulses. Our result on the existence of global attractors is proved for the case where the solutions on the underlying semiflow can possibly be nonunique, and it is even new for the case of solution uniqueness. Additionally, we provide collective versions of the proposed criteria, under which, as we prove, the global attractors exhibit upper-semicontinuity upon perturbation of the problem. The theory is illustrated through several examples of ODEs and PDEs.

Uniform tail-ends estimates of the Navier-Stokes equations on unbounded channel-like domains

This paper deals with the asymptotic compactness of the solutions of the two-dimensional Navier-Stokes equations defined in unbounded channel-like domains. In order to overcome the non-compactness of Sobolev embeddings in unbounded domains, we establish the uniform tail-ends estimates of solutions by showing all the solutions are uniformly small outside a sufficiently large bounded domain for all large time and bounded initial data, which implies the asymptotic compactness and hence the existence of global attractors in the natural energy space. The uniform tail-ends estimates are derived by the scalar stream function of the Navier-Stokes equations which is distinct from the reaction-diffusion equation due to the divergence free condition.

Necessary and sufficient criteria for existence, regularity, and asymptotic stability of enhanced pullback attractors with applications to 3D primitive equations

We introduce several new concepts called enhanced pullback attractors for nonautonomous dynamical systems by improving the compactness and attraction of the usual pullback attractors in strong topology spaces uniformly over some infinite time intervals. Then we establish several necessary and sufficient criteria for the existence, regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which are expected to be applied to many kinds of nonautonomous PDEs. These abstract results are applied to the famous 3D primitive equations modeling the large-scale ocean and atmosphere dynamics. We establish the existence, regularity and asymptotic stability of the enhanced pullback attractors for the nonautonomous primitive equations in [Formula: see text] and [Formula: see text] when the forces satisfy some mild conditions. The uniform pullback asymptotic compactness of the solution operators in [Formula: see text] and [Formula: see text] are established by deriving the uniform “flattening effects” of the solutions in [Formula: see text] and the uniform [Formula: see text]-Hölder continuity of the solutions from [Formula: see text] to [Formula: see text] with respect to the initial data, respectively. Our approaches are applicable to prove the existence of global attractors for the autonomous primitive equations in [Formula: see text] provided [Formula: see text].

Existence of Global Attractor for a 3D Brinkman-Forchheimer Equfation in Some Poincaré Unbounded Domains

In this paper, we study the existence of global attractor of a class of three-dimensional Brinkman-Forchheimer equation in some unbounded domains which satisfies Poincaré inequality. We use the tail estimation method to establish the asymptotic compactness of the solution operator and then prove the existence of the global attractor in [see formula in PDF].

Dynamics of nonlinear Reissner–Mindlin–Timoshenko plate systems

AbstractThe problem of Reissner–Mindlin–Timoshenko plate systems with nonlinear damping terms is considered. The main result is the existence of global attractors. By showing the system is gradient and asymptotic smoothness via a stabilizability inequality, we establish the existence of global attractors with finite fractal dimension. The continuity of global attractors regarding the parameter in a residual dense set is also proved. The above results allow the damping terms with polynomial growth.

Long-time dynamics of a problem of strain gradient porous elastic theory with nonlinear damping and source terms

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of ‘consistency’ due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained.

Upper semicontinuity of the global attractor for Bresse system with second sound

In this paper, we study the long-time dynamics of Bresse system under mixed homogeneous Dirichlet–Neumann boundary conditions. The heat conduction is given by Cattaneo's law. Only the shear angle displacement is damped via the dissipation from the Cattaneo's law, and the vertical displacement and the longitudinal displacement are free. Under quite general assumptions on the source term and based on the semigroup theory, we establish the global well-posedness and the existence of global attractors with finite fractal dimension in natural space energy. Finally, we prove the upper semicontinuous with respect to the relaxation time τ as it converges to zero.

The exponential ordering for nonautonomous delay systems with applications to compartmental Nicholson systems

The exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.

Attractors of Modified Coupled Ginzburg–Landau Model

Bose–Einstein condensation is a gaseous, superfluid state of matter exhibited by bosons as they cool to near absolute zero, which was discovered as early as 1924 but was not experimentally realized until 1995. In 2006, Machida and Koyama developed the corresponding Ginzburg–Landau model for superfluid and Bose–Einstein condensation-spanning phenomena. We mainly consider the global attractor for the initial boundary value problem of the modified coupled Ginzburg–Landau equations, which come from the BCS-BEC crossover model. Combining Gronwall inequality, properties of the binomial function, with some suitable a priori estimates, we establish the existence of global attractors. The attractor results obtained in this paper can provide a strong theoretical basis for the experimental realization of the BCS-BEC spanning phenomenon, and the adopted research method can also serve as a reference for analysing other types of partial differential equation attractors.

Existence and continuity of attractors for a nonlinear piezoelectric beam with Colemam-Gurtin thermal law

In this paper, we study the long-time dynamics of a nonlinear model of a piezoelectric beam subjected to magnetic and thermal effects. The thermal effect acts on the system according to the Coleman-Gurtin law. Using semigroup theory, we prove the existence and uniqueness of solutions. The existence of global attractors is proved by showing that the dynamical system associated with the solutions is quasi-stable and gradient. To conclude, we show that the family of global attractors is continuous on a dense residual set and upper semicontinuous on closed interval.

On a simple model for describing convection of the rotating fluid: Integrability, bifurcations and global dynamics

<p style='text-indent:20px;'>The Glukhovsky-Dolzhansky (GD) model arises naturally from geophysical science, which describes rotating fluid convection inside the ellipsoid. This work aims to provide some new insights into the GD model. (<i>i</i>) We first show that, under some conditions there are homothetic transformations which covert the GD model into other similar quadric physical models, therefore, our results on the GD model can be naturally applied to the investigation of these models. (<i>ii</i>) We propose a complete classification of Darboux polynomials and exponent factors for the GD model, which implies that the GD model has no polynomial, rational, or Darboux first integrals. In addition, some integrable cases of the GD model are also given when the physical parameters are allowed to be non-positive. (<i>iii</i>) The existence of global attractor is proved. The stability and local bifurcations of all co-dimension one and two are investigated. Particularly, we show that the GD model undergoes two dynamical transitions as the Rayleigh number increases. (<i>iv</i>) To understand the asymptotic behavior of the orbits for the GD model, we use the Poincaré compactification method to study its dynamical behavior at infinity. More precisely, we prove that the phase portraits of the GD model at infinity consist of an infinite sequence of periodic solutions and two heteroclinic loops. Our results may help us better understand the complex and rich dynamics of rotating fluid convection.</p>

The periodic initial value problem for Landau–Lifshitz–Bloch–Maxwell systeme

<p style="text-indent:20px;">This paper is devoted to studying the global attractors of the periodic initial value problem for Landau–Lifshitz–Bloch–Maxwell system. First we give the global existence of the smooth solution for this system. Then, we prove the existence of global attractors, the Hausdorff dimension and fractal dimension have been estimated.</p>

Smooth dynamics of a Timoshenko system with hybrid dissipation

In this paper we study the longtime dynamics of a class of thermoelastic Timoshenko beams with history in a nonlinear elastic foundation. Our main result establishes the existence of a global attractor with finite fractal dimension without requiring the so-called equal wave speeds assumption. In addition, the attractor belongs to the phase space of strong solutions. The results are based on properties of gradient systems and a concept of quasi-stability. We believe this is the first study on the existence of global attractors for semilinear Timoshenko systems with hybrid dissipation (heat and memory).

Existence of Global Attractor for a Modified Swift - Hohenberg Equation

This study mainly aims to explore the existence of global attractor for a modified Swift-Hohenberg equation. The method we use was the classical existence theorem of global attractors and the theory of semigroups. Use this method we prove that the equation exist a global attractor in space, and the global attractor attracts whatever bounded subset of in the -norm.

Global attractor for the viscous modified two-component Camassa–Holm equation

We obtain the existence of global attractor for the Cauchy problem of a viscous modified two-component Camassa–Holm equation. The existence of global strong solutions is obtained using Kato’s theory. The key elements in our analysis are the uniform Gronwall lemma and some estimates of the solutions.