Robustness of dynamically gradient multivalued dynamical systems

In this paper we study the robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion studied in J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2965-2984, proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets.Ministerio de Educación, Cultura y DeporteMinisterio de Economía y CompetitividadJunta de AndalucíaFundação de Amparo à Pesquisa do Estado de São PauloConselho Nacional de Desenvolvimento Científico e Tecnológic

On the relationship between solutions of stochastic and random differential inclusions

Some results on the relationship of the solutions of a stochastic di erential inclusion and the corresponding random di erential inclusion obtained after a change of variable are proved. As a consequence, we obtain the pullback convergence of the solutions of the stochastic inclusion to a compact random set. The cases of a reaction-di usion inclusion perturbed by additive and multiplicative noises are considered

Asymptotic Behaviour of Monotone Multi-Valued Dynamical Systems

We introduce the concept of a monotone multi-valued semi- ow as an order-preserving map. This de nition is motivated by the applications in the theory of di¤erential equations without uniqueness of solutions. For an order preserving multi-valued semi- ow we prove several results on the structure of the global attractor. Some applications to models governed by ordinary di¤erential equations and delay equations with continuous right-hand side are presented. In particular, the abstract results are applied to a biochemical control circuit

The dimension of attractors of nonautonomous partial differential equations

The concept of nonautonomous (or cocycle) attractor has become a proper tool for the study of the asymptotic behaviour of general nonautonomous partial differential equations. This is a time-dependent family of compact sets, invariant for the associated process and attracting “from ¡1”: In general, the concept is rather different from the classical one of global attractor for autonomous dynamical systems. We prove a general result on the finite fractal dimensionality of each compact set of this family. In this way, we generalize previous results of Chepyzhov and Vishik in [6]. Our results are also applied to differential equations with a nonlinear term having polynomial growth at most

Stabilisation of differential inclusions and PDEs without uniqueness by noise

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 Itˆo 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

Global Attractors for Multivalued Random Dynamical Systems

We introduce the concept of multivalued random dynamical system (MRDS) as a measurable multivalued flow satisfying the cocycle property. We show how this is a suitable framework for the study of the asymptotic behaviour of some multivalued stochastic parabolic equations by generalizing the concept of global random attractor to the case of a MRDS

Global attractors for multivalued random semiflows generated by random differential inclusions with additive noise

We introduce the concept of multivalued random dynamical system (MRDS) as a measurable multivalued flow satisfying the cocycle property. We show how this is a suitable framework for the study of the asymptotic behaviour of some multivalued stochastic parabolic equations by generalizing the concept of global random attractor to the case of a MRDS

Approximation of attractors for multivalued random dynamical systems

The concept of global attractor for stochastic partial differential inclusions has been recently introduced as a joint generalization of the theory of random attractors for random dynamical systems and global attractors for multivalued semiflows. We present a general result on the upper semicontinuity of attractors for multivalued random dynamical systems. In particular, our theory shows how the random attractor associated to a small random perturbation of a (deterministic) partial differential inclusion approximates the global attractor of the limiting problem. Some applications ilustrate the results

Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term

In this paper we establish a strong comparison principle for a nonautonomous differential inclusion with a forcing term of Heaviside type. Using this principle, we study the structure of the global attractor in both the autonomous and nonautonomous cases. In particular, in the last case we prove that the pullback attractor is confined between two special bounded complete trajectories, which play the role of nonautonomous equilibria