DCDS
The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations
Peter E. Kloeden José Valero
The Kneser theorem for ordinary differential equations without uniqueness says that the attainability set is compact and connected at each instant of time. We establish corresponding results for the attainability set of weak solutions for the 3D Navier-Stokes equations satisfying an energy inequality. First, we present a simplified proof of our earlier result with respect to the weak topology in the space $H$. Then we prove that this result also holds with respect to the strong topology on $H$ provided that the weak solutions satisfying the weak version of the energy inequality are continuous. Finally, using these results, we show the connectedness of the global attractor of a family of setvalued semiflows generated by the weak solutions of the NSE satisfying suitable properties.
keywords: weak connectedness Attainability set Navier-Stokes equations global weak attractor. globally modified Navier-Stokes Equations weak compactness Kneser property
DCDS
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
DCDS
Non--autonomous and random attractors for delay random semilinear equations without uniqueness
Tomás Caraballo M. J. Garrido-Atienza B. Schmalfuss José Valero
We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by $\mathbb{R}^{-}$. In particular, we do not assume the uniqueness of solutions of these equations.
keywords: multi-valued non-autonomous and random dynamical systems pullback and random attractors delay differential equations. Cocycles
DCDS-B
Global attractors for $p$-Laplacian differential inclusions in unbounded domains
Jacson Simsen José Valero
In this work we consider a differential inclusion governed by a p-Laplacian operator with a diffusion coefficient depending on a parameter in which the space variable belongs to an unbounded domain. We prove the existence of a global attractor and show that the family of attractors behaves upper semicontinuously with respect to the diffusion parameter. Both autonomous and nonautonomous cases are studied.
keywords: upper semicontinuity. differential inclusions Unbounded domains $p$-Laplacian operator attractors
DCDS
On dimension of attractors of differential inclusions and reaction-diffussion equations
Francisco Balibrea José Valero
In this paper we improve a general theorem of O.A. Ladyzhenskaya on the dimension of compact invariant sets in Hilbert spaces. Then we use this result to prove that the Hausdorff and fractal dimensions of global compact attractors of differential inclusions and reaction-diffusion equations are finite.
keywords: Attractor Hausdorff dimension fractal dimension.
DCDS-B
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions
Tomás Caraballo María J. Garrido–Atienza Björn Schmalfuss José Valero
In this work we present the existence and uniqueness of pullback and random attractors for stochastic evolution equations with infinite delays when the uniqueness of solutions for these equations is not required. Our results are obtained by means of the theory of set-valued random dynamical systems and their conjugation properties.
keywords: Multivalued non-autonomous and random dynamical systems functional stochastic equations conjugacy method. pullback and random attractors
DCDS-B
Attractors for a random evolution equation with infinite memory: Theoretical results
Tomás Caraballo María J. Garrido-Atienza Björn Schmalfuss José Valero

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

keywords: Pullback and random attractor random dynamical system random delay equation infinite delay
DCDS-B
Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions
María Anguiano Tomás Caraballo José Real José Valero
The existence of a pullback attractor for a reaction-diffusion equations in an unbounded domain containing a non-autonomous forcing term taking values in the space $H^{-1}$, and with a continuous nonlinearity which does not ensure uniqueness of solutions, is proved in this paper. The theory of set-valued non-autonomous dynamical systems is applied to the problem.
keywords: asymptotic compactness multivalued evolution process Pullback attractor non-autonomous reaction-diffusion equation.
DCDS-B
Attractors for a non-linear parabolic equation modelling suspension flows
José M. Amigó Isabelle Catto Ángel Giménez José Valero
In this paper we prove the existence of a global attractor with respect to the weak topology of a suitable Banach space for a parabolic scalar differential equation describing a non-Newtonian flow. More precisely, we study a model proposed by Hébraud and Lequeux for concentrated suspensions.
keywords: global attractor set-valued dynamical system Non-Newtonian fluids
DCDS
Asymptotic behaviour of a logistic lattice system
Tomás Caraballo Francisco Morillas José Valero
In this paper we study the asymptotic behaviour of solutions of a lattice dynamical system of a logistic type. Namely, we study a system of infinite ordinary differential equations which can be obtained after the spatial discretization of a logistic equation with diffusion. We prove that a global attractor exists in suitable weighted spaces of sequences.
keywords: logistic equation Lattice dynamical systems population models. global attractor ordinary differential equations

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