Tomás Caraballo

Upper semicontinuity of attractors for small random perturbations of dynamical systems

The relationship between random attractors and global attractors for dynamical systems is studied. If a partial differential equation is perturbed by an E-small random term and certain hypotheses are satisfied, the upper semicontinuity of the random attractors is obtalned as c goes to zero. The results are applied to the Navier-Stokes equations and a problem of reaction-diffusion type, both perturbed by an additive white noise.

Pullback Attractors of Nonautonomous and Stochastic Multivalued Dynamical Systems

In this paper we study the existence of pullback global attractors for multivalued processes generated by differential inclusions. First, we define multivalued dynamical processes, prove abstract results on the existence of ω-limit sets and global attractors, and study their topological properties (compactness, connectedness). Further, we apply the abstract results to nonautonomous differential inclusions of the reaction–diffusion type in which the forcing term can grow polynomially in time, and to stochastic differential inclusions as well.

Exponential stability of mild solutions of stochastic partial differential equations with delays

A semiiinear stochastic partial differential equation with variable delays is considered. Sufficient conditions for the exponential stability in the p-th mean of mild solutions are obtained. Also, pathwise exponential stability is proved. Since the technique ofLyapunov functions is not suitable for delayed equations, the results have been proved by using the properties of the stochastic convolution. As the sufficient conditions obtained are also valid for the case without delays, one can ensure exponential stability of mild solution in some cases where the sufficient conditions in Ichikawa [11] do not give any answer. The results are illustrated with some examples

A Stochastic Pitchfork Bifurcation in a Reaction-Diffusion Equation

We study in some detail the structure of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise, du=( Δu+βu- u 3 ) dt+σuod W t ,x∈D⊂ R m First we prove, for m ⩽ 5, a lower bound on the dimension of the random attractor, which is of the same order in β as the upper bound we derived in an earlier paper, and is the same as that obtained in the deterministic case. Then we show, for m = 1, that as β passes through λ1 (the first eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the first example of such a stochastic bifurcation in an infinite–dimensional setting. Central to our approach is the existence of a random unstable manifold.

RECENT DEVELOPMENTS IN DYNAMICAL SYSTEMS: THREE PERSPECTIVES

This paper aims to an present account of some problems considered in the past years in Dynamical Systems, new research directions and also provide some open problems.

Navier-Stokes Equations with Delays

Some results on the existence and uniqueness of solutions to Navier–;Stokes equations when the external force contains some hereditary characteristics are proved.

Asymptotic behaviour of two–dimensional Navier–Stokes equations with delays

Some results on the asymptotic behaviour of solutions to Navier–Stokes equations when the external force contains some hereditary characteristics are proved. We show two different approaches to prove the convergence of solutions to the stationary one, when this is unique. The first is a direct method, while the second is based on a Razumikhin–type method.The Navier-Stokes equations govern the motion of usual fluids like water, air, oil, etc. These equations have been the subject of numerous works since the first paper of Leray was published in 1933 (see Constantin & Foias 1988; Lions 1969; Temam 1979, and the references therein). In our recent work Caraballo & Real (2001) we consider a Navier-Stokes model in which the external force contains some hereditary features and prove the existence of weak solutions. These situations containing delays may appear when we want to control the system (in certain sense) by applying a force which takes into account not only the present state of the system but the history of the solution. Another interesting problem concerns the asymptotic behaviour of the systems, since this analysis can provide useful information on the future evolution of the system. This will be the main aim of this paper. To this end, let Ω ⊂ R be an open bounded set with regular boundary Γ, and consider the following functional 2D−Navier-Stokes problem (for further details and notations see Lions 1969 and Temam 1979):    ∂u

Asymptotic Exponential Stability of Stochastic Partial Differential Equations with Delay

Sufficient conditions for pathwise asymptotic exponential stability of the solution of the stochastic PDE with delay d x t = Ax tdt + B(xp(t)) dwt are given. The assumptions on the operators A and B are essentially the same as in the case without delay. In addition, our deduction also shows an alternative proof for some of the results in this case. In fact, the crucial difference is that we do not use the operator P employed by Haussmann and Ichikawa.

Pullback and Forward Attractors for a Damped Wave Equation with Delays

The existence of a pullback (and also a uniform forward) attractor is proved for a damped wave equation containing a delay forcing term which, in particular, covers the models of sine–Gordon type. The result follows from the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the model.

A Comparison between Two Theories for Multi-Valued Semiflows and Their Asymptotic Behaviour

This paper presents a comparison between two abstract frameworks in which one can treat multi-valued semiflows and their asymptotic behaviour. We compare the theory developed by Ball (1997) to treat equations whose solutions may not be unique, and that due to Melnik and Valero (1998) tailored more for differential inclusions. Although they deal with different problems, the main ideas seem quite similar. We study their relationship in detail and point out some essential technical problems in trying to apply Balls theory to differential inclusions.