Attractors for Multivalued Impulsive Systems: Existence and Applications to Reaction-Diffusion System

Abstract

In this paper, we develop a general approach to investigate limit dynamics of infinite-dimensional dissipative impulsive systems whose initial conditions do not uniquely determine their long time behavior. Based on the notion of an uniform attractor, we show how to describe limit behavior of such complex systems with the help of properties of their components. More precisely, we prove the existence of the uniform attractor for an impulsive multivalued system in terms of properties of nonimpulsive semiflow and impulsive parameters. We also give an application of these abstract results to the impulsive reaction-diffusion system without uniqueness.

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

“…Following the work in [48], we describe the general construction of the impulsive dynamical system. Suppose that a continuous semigroup G : R + × X → X is given on the phase space X.…”

Dynamics of Nonautomous Impulsive Multivalued Processes

“…There are also several results regarding the stability and control theory of these systems, see for example [25][26][27][28][29] In this paper we focus on the study of the dynamics of systems in the nonautonomous and impulsive multivalued situation. For example, in [23,30,31] they consider autonomous multivalued dynamical systems. We study the nonautonomous situation and we give conditions in order to guarantee the existence of the pullback attractor.…”