Morse decomposition of global attractors with infinite components
Tomás Caraballo Juan C. Jara José A. Langa José Valero
Discrete & Continuous Dynamical Systems - A 2015, 35(7): 2845-2861 doi: 10.3934/dcds.2015.35.2845
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
Discrete & Continuous Dynamical Systems - B 2016, 21(9): 2949-2967 doi: 10.3934/dcdsb.2016081
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
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 483-497 doi: 10.3934/dcds.2007.18.483
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
Discrete & Continuous Dynamical Systems - A 2000, 6(4): 875-892 doi: 10.3934/dcds.2000.6.875
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
Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing
Hongyong Cui Mirelson M. Freitas José A. Langa
Discrete & Continuous Dynamical Systems - B 2018, 23(3): 1297-1324 doi: 10.3934/dcdsb.2018152

In this paper, we study the squeezing property and finite dimensionality of cocycle attractors for non-autonomous dynamical systems (NRDS). We show that the generalized random cocycle squeezing property (RCSP) is a sufficient condition to prove a determining modes result and the finite dimensionality of invariant non-autonomous random sets, where the upper bound of the dimension is uniform for all components of the invariant set. We also prove that the RCSP can imply the pullback flattening property in uniformly convex Banach space so that could also contribute to establish the asymptotic compactness of the system. The cocycle attractor for 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing is studied as an application.

keywords: Non-autonomous random dynamical system random cocycle attractor finite dimensionality squeezing property 2D stochastic Navier-Stokes equations
Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations
P.E. Kloeden José A. Langa José Real
Communications on Pure & Applied Analysis 2007, 6(4): 937-955 doi: 10.3934/cpaa.2007.6.937
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
Discrete & Continuous Dynamical Systems - B 2015, 20(3): i-ii doi: 10.3934/dcdsb.2015.20.3i
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
Communications on Pure & Applied Analysis 2008, 7(6): 1375-1392 doi: 10.3934/cpaa.2008.7.1375
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.
On random cocycle attractors with autonomous attraction universes
Hongyong Cui Mirelson M. Freitas José A. Langa
Discrete & Continuous Dynamical Systems - B 2017, 22(9): 3379-3407 doi: 10.3934/dcdsb.2017142

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
Pullback exponential attractors
José A. Langa Alain Miranville José Real
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1329-1357 doi: 10.3934/dcds.2010.26.1329
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

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