Weak pullback attractors of setvalued processes

Weak pullback attractors are defined for nonautonomous setvalued processes and their existence and upper semicontinuous convergence under perturbation is established. Unlike strong pullback attractors, invariance and pullback attraction here are required only for at least one trajectory rather than all trajectories at each starting point. The concept is useful in, for example, continuous time control systems and is related to that of viability.     

Pullback attractors for asymptotically compact non-autonomous dynamical systems

First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.     

Existence of backwards-compact pullback attractors for non-autonomous lattice dynamical systems

A pullback attractor is called backward compact if the union of attractors over the past time is pre-compact. We show that this kind of attractor exists for the first-order non-autonomous lattice dynamical system when the external force is backwards tempered and backwards asymptotically tail-null.     

Upper semicontinuity of pullback attractors for multi-valued random cocycle

In this paper we study the upper semicontinuity of random attractors for multi-valued random cocycle when small random perturbations approach zero or small perturbation for random cocycle is considered. Furthermore, we consider the upper semicontinuity of random attractors for multi-valued random cocycle under the condition which the metric dynamical systems is ergodic.     

Regularity of pullback attractors for non-autonomous stochastic coupled reaction-diffusion systems

We consider the dynamical behavior of the typical non-autonomous autocatalytic stochastic coupled reaction-diffusion systems on the entire space $\mathbb{R}^n$. Some new uniform asymptotic estimates are implemented to investigate the existence of pullback attractors in the Sobolev space $H^1(\mathbb{R}^n)^3$ for the three-component reversible Gray-Scott system.     

Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations

A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.     

Double stabilities of pullback random attractors for stochastic delayed p-Laplacian equations

We provide a method to study the double stabilities of a pullback random attractor (PRA) generated from a stochastic partial differential equation (PDE) with delays, such a PRA is actually a family of compact random sets A_ϱ(t,·), where t is the current time and ϱ is the memory time. We study its longtime stability, which means the attractor semiconverges to a compact set as the current time tends to minus infinity, and also its zero-memory stability, which means the delayed attractor semiconverges to the nondelayed attractor as the memory time tends to zero. The stochastic nonautonomous p-Laplacian equation with variable delays on an unbounded domain will be applied to illustrate the method and some suitable assumptions about the nonlinearity and time-dependent delayed forces can ensure existence, backward compactness, and double stabilities of a PRA.     

Upper semicontinuous dependence of pullback attractors on time scales

Dynamical equations on time scales typically generate a nonautonomous process, even when the vector field function does not depend explicitly on time. Nonautonomous pullback attractors are thus the appropriate generalisation of autonomous attractors to time scale dynamics. The existence of a pullback attractor follows when the process has a pullback absorbing set. Assuming that a dynamical equation over a given time scale which has no rapidly increasing gaps satisfies a certain dissipativity condition, and thus possesses a pullback attractor, and that its solutions depend uniformly on initial data including the time scale, it is shown that the same dynamical equation over nearby time scales also has a pullback attractor, whose component sets converge upper semicontinuously to the corresponding component sets of the pullback attractor of the original system.     

On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems

For an abstract dynamical system, we establish, under minimal assumptions, the existence of D-attractor, i.e. a pullback attractor for a given class D of families of time varying subsets of the phase space. We relate this concept with the usual attractor of fixed bounded sets, pointing out its usefulness in order to ensure the existence of this last attractor in particular situations. Moreover, we prove that under a simple assumption these two notions of attractors generate, in fact, the same object. This is then applied to a Navier-Stokes model, improving some previous results on attractor theory.     

Recession cones and asymptotically compact sets

The present note is concerned with the study of the relations between the notions of asymptotic cones introduced by Dedieu and that of recession cones introduced by Luc. Conditions under which these notions coincide are given, as well as the fact that the compactness condition used by Luc is related (more restrictively) to asymptotic compactness. As an application of these notions, a result on proper efficiency in the sense of Lampe, established by Luc in finite dimensions, is extended to the infinite-dimensional case.This work was partially done while the author was visiting the Department of Mathematics of the University of Pisa, Pisa, Italy.     

Regularity of pullback attractors and random equilibrium for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains

This paper is concerned with the stochastic Fitzhugh-Nagumo system with non-autonomous terms as well as Wiener type multiplicative noises. By using the so-called notions of uniform absorption and uniformly pullback asymptotic compactness, the existences and upper semi-continuity of pullback attractors are proved for the generated random cocycle in $L^l(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$ for any $l\in(2,p]$. The asymptotic compactness of the first component of the system in $L^p(\mathbb{R}^N)$ is proved by a new asymptotic a priori estimate technique, by which the plus or minus sign of the nonlinearity at large values is not required. Moreover, the condition on the existence of the unique random fixed point is obtained, in which case the influence of physical parameters on the attractors is analysed.     

Uniform compact attractors for the coupled suspension bridge equations

In this paper, we investigate the uniformly absorbing set for the coupled suspension bridge equations with a new class of external forces, and then prove the existence of the uniform attractors for the family of process corresponding to the equations.     

一类渐近线性Neumann问题正解的存在性

利用变分法获得了一类渐近线性N eum ann问题正解的存在性结果.     

Existence of global attractors for a nonlinear wave equation

Using a new method developed in [Q. Ma, S. Wang, C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Math. J. 5 (6) (2002) 1542–1558], we prove the existence of the global attractor for a nonlinear wave equation.     

Existence of solution of the pullback equation involving volume forms

Let Ω ⊂ ℝ ⁿ be a smooth, bounded domain. We study the existence and regularity of diffeomorphisms of Ω satisfying the volume form equation