Morse decomposition of global attractors with infinite components
Tomás Caraballo Juan C. Jara José A. Langa José Valero
In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.
keywords: Morse decomposition infinite components gradient dynamics gradient-like semigroup. Lyapunov function
Equi-attraction and continuity of attractors for skew-product semiflows
Tomás Caraballo Alexandre N. Carvalho Henrique B. da Costa José A. Langa
In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated to non-autonomous dynamical systems. To this aim proper notions of equi-attraction have to be introduced in phase spaces where the driving systems depend on a parameter. Results on the upper and lower-semicontinuity of uniform and cocycle attractors are relatively new in the literature, as a deep understanding of the internal structure of these sets is needed, which is generically difficult to obtain. The notion of lifted invariance for uniform attractors allows us to compare the three types of attractors and introduce a common framework in which to study equi-attraction and continuity of attractors. We also include some results on the rate of attraction to the associated attractors.
keywords: skew-product semiflows. Equi-attraction continuity of attractors
Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations
José A. Langa James C. Robinson Aníbal Rodríguez-Bernal A. Suárez A. Vidal-López
The goal of this work is to study the forward dynamics of positive solutions for the non-autonomous logistic equation $u_{t}-\Delta u=\lambda u-b(t)u^{p}$, with $p>1$, $b(t)>0$, for all $t\in \mathbb{R}$, $\lim_{t\to \infty }b(t)=0$. While the pullback asymptotic behaviour for this equation is now well understood, several different possibilities are realized in the forward asymptotic regime.
keywords: Unbounded forwards attractor Non-autonomous reaction diffusion equations.
Stability and random attractors for a reaction-diffusion equation with multiplicative noise
Tomás Caraballo José A. Langa James C. Robinson
We study the asymptotic behaviour of a reaction-diffusion equation, and prove that the addition of multiplicative white noise (in the sense of Itô) stabilizes the stationary solution $x\equiv 0$. We show in addition that this stochastic equation has a finite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario.
keywords: Random attractors Chafee-Infante equation. stochastic stabilization Hausdorff dimension
Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations
P.E. Kloeden José A. Langa José Real
The existence and finite fractal dimension of a pullback attractor in the space $V$ for a three dimensional system of the nonautonomous Globally Modified Navier-Stokes Equations on a bounded domain is established under appropriate properties on the time dependent forcing term. These equations were proposed recently by Caraballo et al and are obtained from the Navier- Stokes Equations by a global modification of the nonlinear advection term. The existence of the attractor is obtained via the flattening property, which is verified.
keywords: weak solutions flattening property existence and uniqueness of strong solutions 3-dimensional Navier-Stokes equations nonautonomous and pullback attractors.
Alexandre N. Carvalho José A. Langa James C. Robinson
We were very pleased to be given the opportunity by Prof. Peter Kloeden to edit this special issue of Discrete and Continuous Dynamical Systems - Series B on the asymptotic dynamics of non-autonomous systems.

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Stabilisation of differential inclusions and PDEs without uniqueness by noise
Tomás Caraballo José A. Langa José Valero
We prove that the asymptotic behaviour of partial differential inclusions and partial differential equations without uniqueness of solutions can be stabilised by adding some suitable Itô noise as an external perturbation. We show how the theory previously developed for the single-valued cases can be successfully applied to handle these set-valued cases. The theory of random dynamical systems is used as an appropriate tool to solve the problem.
keywords: Multivalued semiflows stabilization by Ito's noise random dynamical systems.
Pullback exponential attractors
José A. Langa Alain Miranville José Real
In this work, we show how to construct a pullback exponential attractor associated with an infinite dimensional dynamical system, i.e., a family of time dependent compact sets, with finite fractal dimension, which are positively invariant and exponentially attract in the pullback sense every bounded set of the phase space. Our construction is based on the one in Efendiev et al. [11] in which a uniform forwards (and so also pullback) exponential attractor is constructed. We relax the conditions in [11] in order to obtain an unbounded family of exponential attractors for which the uniform convergence fails so that only the pullback attraction is expected. Thus, by proving that global pullback attractors are included in our family of exponential attractors, we generalize the concept of an exponential attractor to the theory of infinite dimensional non-autonomous dynamical systems. We illustrate our results on a 2D Navier-Stokes system in bounded domains.
keywords: 2D Navier-Stokes equations. non-autonomous PDEs pullback attractors infinite-dimensional dynamical systems exponential attractors
Non-autonomous dynamical systems
Alexandre N. Carvalho José A. Langa James C. Robinson
This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors. All of them are closely related to the study of the pullback asymptotic limits of the dynamical system, from which naturally emerges the concept of a pullback attractor. In the first part of this paper we want to clarify these different dynamical scenarios and the relations between their corresponding attractors.
    If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.
keywords: Morse decomposition pullback attractor Skew-product semiflow uniform attractor. cocycle attractor Morse--Smale systems
On random cocycle attractors with autonomous attraction universes
Hongyong Cui Mirelson M. Freitas José A. Langa

In this paper, for non-autonomous RDS we study cocycle attractors with autonomous attraction universes, i.e. pullback attracting some autonomous random sets, instead of non-autonomous ones. We first compare cocycle attractors with autonomous and non-autonomous attraction universes, and then for autonomous ones we establish some existence criteria and characterization. We also study for cocycle attractors the continuity of sections indexed by non-autonomous symbols to find that the upper semi-continuity is equivalent to uniform compactness of the attractor, while the lower semi-continuity is equivalent to an equi-attracting property under some conditions. Finally, we apply these theoretical results to 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing.

keywords: Random cocycle attractor pullback attractor basis of attraction 2D stochastic Navier-Stokes equation

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