José Valero

On Attractors of Multivalued Semi-Flows and Differential Inclusions

In this paper we study the existence of global attractors for multivalued dynamical systems. These theorems are then applied to dynamical systems generated by differential inclusions for which the solution is not unique for a given initial state. Finally, some boundary-value problems are considered.

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.

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.

Attractors of Parabolic Equations Without Uniqueness

In this paper we study the existence of global compact attractors for nonlinear parabolic equations of the reaction-diffusion type and variational inequalities. The studied equations are generated by a difference of subdifferential maps and are not assumed to have a unique solution for each initial state. Applications are given to inclusions modeling combustion in porous media and processes of transmission of electrical impulses in nerve axons.

Attractors of Multivalued Dynamical Processes Generated by Phase-Field Equations

In this paper we define and study multivalued dynamical processes in Hausdorff topological spaces. Existence theorems for attractors of multivalued processes are proved, their topological properties are studied. The abstract results are applied to a system of phase-field equations without conditions providing uniqueness of solutions and to nonautonomous differential inclusions.

Attractors of multivalued semiflows generated by differential inclusions and their approximations

We prove the existence of global compact attractors for differential inclusions and obtain some results concerning the continuity and upper semicontinuity of the attractors for approximating and perturbed inclusions. Applications are given to a model of regional economic growth.

On Global Attractors of Multivalued Semiprocesses and Nonautonomous Evolution Inclusions

In the first chapter, we considered the existence and properties of global attractors for autonomous multivalued dynamical systems. When the equation is nonautonomous, new and challenging difficulties appear. In this case, if uniqueness of the Cauchy problem holds, then the usual semigroup of operators becomes a two-parameter semigroup or process [38, 39], as we have to take into account the initial and the final time of the solutions.

On the Kneser property for the complex Ginzburg–Landau equation and the Lotka–Volterra system with diffusion

As we have seen in the previous chapters when we consider the Cauchy problem of a differential equation and uniqueness fails to hold (or it is not known to hold), then we have to consider a set of solutions corresponding to a given initial data.

Pullback attractors for a class of extremal solutions of the 3D Navier–Stokes system

The study of the asymptotic behavior of the weak solutions of the three-dimensional (3D for short) Navier–Stokes system is a challenging problem which is still far to be solved in a satisfactory way. In particular, the existence of a global attractor in the strong topology is an open problem for which only some partial or conditional results are given (see [3, 4, 6, 15, 17, 19, 20, 27, 38]). Concerning the existence of trajectory attractors, some results are proved in [13, 18, 36]. The main difficulty in this problem (but not the only one!) is to prove the asymptotic compactness of solutions (see [2], for a review on these questions).

COMPARISON BETWEEN TRAJECTORY AND GLOBAL ATTRACTORS FOR EVOLUTION SYSTEMS WITHOUT UNIQUENESS OF SOLUTIONS

In this paper we make a thorough comparison between the theory of global attractors for multivalued semiflows and the theory of trajectory attractors, two methods which are useful for studying the asymptotic behavior of solution for equations without uniqueness of the Cauchy problem. We show that under some conditions the formula A = U(0) takes place for the global attractor A of a multivalued semiflow and the trajectory attractor U of the associated translation semigroup. We apply these results to reaction–diffusion equations and hyperbolic equations, obtaining also new theorems concerning the existence of related trajectory attractors.