DCDS

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.

DCDS

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.

DCDS

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.

DCDS-B

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.

DCDS

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.

DCDS-B

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.

DCDS-B

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.

DCDS-B

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.

DCDS-B

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.

DCDS-B

In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero.