About the Structure of Attractors for a Nonlocal Chafee-Infante Problem

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.

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.

PEANO'S THEOREM AND ATTRACTORS FOR LATTICE DYNAMICAL SYSTEMS

In this paper, we prove the existence of solutions for first order lattice dynamical systems with continuous nonlinear term obtained via discretization of a reaction–diffusion system. Since the uniqueness of the Cauchy problem is not guaranteed, we define a multivalued semiflow and prove the existence of a global compact attractor.