Uniform attractors for non-autonomous Klein-Gordon-Schrǒdinger lattice systems

The existence of a compact uniform attractor for a family of processes corresponding to the dissipative non-autonomous Klein-Gordon-Schrodinger lattice dynamical system is proved. An upper bound of the Kolmogorov entropy of the compact uniform attractor is obtained, and an upper semicontinuity of the compact uniform attractor is established.     

Uniform attractors for non-autonomous Klein-Gordon-Schr/Sdinger lattice systems

The existence of a compact uniform attractor for a family of processes corre- sponding to the dissipative non-autonomous Klein-Gordon-SchrSdinger lattice dynamical system is proved. An upper bound of the Kolmogorov entropy of the compact uniform attractor is obtained, and an upper semicontinuity of the compact uniform attractor is established.     

ATTRACTORS OF NONAUTONOMOUS SCHR?DINGER EQUATIONS

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Random attractors for asymptotically upper semicompact multivalue random semiflows

The present paper studied the dynamics of some multivalued random semi- flow.The corresponding concept of random attractor for this case was introduced to study asymptotic behavior.The existence of random attractor of multivalued random semiflow was proved under the assumption of pullback asymptotically upper semicompact,and this random attractor is random compact and invariant.Furthermore,if the system has ergodicity,then this random attractor is the limit set of a deterministic bounded set.     

Random Dynamics of Stochastic Reaction–Diffusion Systems with Additive Noise

For a typical autocatalytic stochastic reaction–diffusion system with additive noises, the multicomponent reversible Gray–Scott reaction–diffusion system on a two-dimensional bounded domain, the existence of a random attractor and its attracting regularity are proved through the sharp uniform estimates showing respectively the pullback absorbing, asymptotically compact, and flattening properties.     

Dissipation and Compact Attractors

We present an approach to the study of the qualitative theory of infinite dimensional dynamical systems. In finite dimensions, most of the success has been with the discussion of dynamics on sets which are invariant and compact. In the infinite dimensional case, the appropriate setting is to consider the dynamics on the maximal compact invariant set. In dissipative systems, this corresponds to the compact global attractor. Most of the time is devoted to necessary and sufficient conditons for the existence of the compact global attractor. Several important applications are given as well as important results on the qualitative properties of the flow on the attractor.     

Pullback Attractors in Dissipative Nonautonomous Differential Equations Under Discretization

The effects of discretization on the nonautonomous pullback attractors of skew-product flows generated by a class of dissipative differential equations, are investigated, It is assumed that the vector, field of the differential equations varies in time due to the input of an autonomous dynamical system acting on a compact metric space. In particular, it is shown that the corresponding discrete time skew-product system generated by a one-step numerical scheme with variable timesteps also has a pullback attractor, the component subsets of which converge upper semicontinuously to their counterparts of the pullback attractor of the original continuous time system.     

Finite Dimensional Attractor for a One-Phase Stefan Problem with Kinetics

For a one-phase free-boundary problem with kinetics, which is known to generate a rich dynamics, we study evolution of the infinitesimal volume along the trajectories in the attractor. We demonstrate that for sufficiently large m that is defined solely by the properties of the kinetics function the m-dimensional volume decays exponentially. This property combined with the uniform differentiability of the semigroup leads to the conclusion that the Hausdorff dimension of the attractor is finite.     

Properties of the attractor of a scalar parabolic PDE

We consider a general scalar one-dimensional semilinear parabolic partial differential equation generating a semiflow with an attractor in an adequate state space. Generalizing known results, it is shown that this attractor is the graph of a function over a compact subset of a finite-dimensional subspace of the state space. In addition, we construct an example with a special interest for the geometric or bifurcation theory of this type of parabolic equations.     

ATTRACTORS OF NONAUTONOMOUS SCHRODINGER EQUATIONS

The long-time behaviour of a two-dimensional nonautonomous nonlinear SchrOdinger equation is considered. The existence of uniform attractor is proved and the up per bound of the uniform attractor' s Hausdorff dimension is given.     

GLOBAL ATTRACTOR FOR HASEGAWA-MIMA EQUATION

The long time behavior of solution of the Hasegawa-Mima equation with dissipation term was considered. The global attractor problem of the Hasegawa-Mima equation with initial periodic boundary condition was studied. Applying the uniform a priori estimates method, the existence of global attractor of this problem was proved, and also the dimensions of the global attractor was estimated.     

The attractor of the scalar reaction diffusion equation is a smooth graph

For the scalar reaction diffusion equation with Dirichlet boundary conditions, it is proved that its maximal compact attractor is the graph of a C¹ function from a subset with nonempty interior of a subspace of the state space the dimension of which is equal to the maximal Morse index of the equilibria of the equation.     

Global Attractors of Evolutionary Systems

An abstract framework for studying the asymptotic behavior of a dissipative evolutionary system with respect to weak and strong topologies was introduced in Cheskidov and Foias (J Differ Equ 231:714–754, 2006) primarily to study the long-time behavior of the 3D Navier-Stokes equations (NSE) for which the existence of a semigroup of solution operators is not known. Each evolutionary system possesses a global attractor in the weak topology, but does not necessarily in the strong topology. In this paper we study the structure of a global attractor for an abstract evolutionary system, focusing on omega-limits and attracting, invariant, and quasi-invariant sets. We obtain weak and strong uniform tracking properties of omega-limits and global attractors. In addition, we discuss a trajectory attractor for an evolutionary system and derive a condition under which the convergence to the trajectory attractor is strong.     

Doubly nonlinear parabolic-type equations as dynamical systems

In this paper, we study a class of doubly nonlinear parabolic PDEs, where, in addition to some weak nonlinearities, also some mild nonlinearities of porous media type are allowed inside the time derivative. In order to formulate the equations as dynamical systems, some existence and uniqueness results are proved. Then the existence of a compact attractor is shown for a class of nonlinear PDEs that include doubly nonlinear porous medium-type equations. Under stronger smoothness assumptions on the nonlinearities, the finiteness of the fractal dimension of the attractor is also obtained.     

Global attractor for Klein-Gordon-Schroedinger lattice system

We considered the longtime behavior of solutions of a coupled lattice dynamical system of Klein-Gordon-Schroedinger equation （KGS lattice system）. We first proved the existence of a global attractor for the system considered here by introducing an equivalent norm and using ＂End Tails＂ of solutions. Then we estimated the upper bound of the Kolmogorov delta-entropy of the global attractor by applying element decomposition and the covering property of a polyhedron by balls of radii delta in the finite dimensional space. Finally, we presented an approximation to the global attractor by the global attractors of finite-dimensional ordinary differential systems.     

Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations

In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.     

Global Attractor for Impulsive Reaction-Diffusion Equation

In this paper, we consider a reaction-diffusion equation with nonsmooth nonlinearity whose solutions have impulse effects at fixed moments of time. We show how this object generates a nonautonomous multivalued dynamical system and prove the existence of a compact semiinvariant global attractor in the phase space. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 3, pp. 319–328, July–September, 2005.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

Global attractor for Klein-Gordon-Schrodinger lattice system

We considered the longtime behavior of solutions of a coupled lattice dynamical system of Klein-Gordon-Schrodinger equation (KGS lattice system). We first proved the existence of a global attractor for the system considered here by introducing an equivalent norm and using "End Tails" of solutions. Then we estimated the upper bound of the Kolmogorov delta-entropy of the global attractor by applying element decomposition and the covering property of a polyhedron by balls of radii delta in the finite dimensional space. Finally, we presented an approximation to the global attractor by the global attractors of finite-dimensional ordinary differential systems.     

Random Attractor Associated with the Quasi-Geostrophic Equation

We study the long time behavior of the solutions to the 2D stochastic quasi-geostrophic equation on \({\mathbb {T}}^2\) driven by additive noise and real linear multiplicative noise in the subcritical case (i.e. \(\alpha >\frac{1}{2}\)) by proving the existence of a random attractor. The key point for the proof is the exponential decay of the \(L^p\)-norm and a boot-strapping argument. The upper semicontinuity of random attractors is also established. Moreover, if the viscosity constant is large enough, the system has a trivial random attractor.