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.
Highlights
Another approach was developed in [35,36,37,38] and uses the notion of the uniform attractor
Let us note that this paper uses a common approach applied to study attracting sets of impulsive infinite-dimensional systems, as, for example, in [31,32,33,34,35,36,37,38,39]. In these and many other works, the existence of a uniform attractor is proved by means of asymptotic compactness verification for the impulsive semiflow. is verification is rather difficult already in case of relatively simple problems because the times of impulsive actions are usually not known in advance. e novelty of our results is that they provide new sufficient conditions for the asymptotic compactness property which can be verified. is allows us to consider a more general class of systems with more general impulsive action classes
We were able to establish the existence of a uniform attractor for an impulsive system of N ∈ N reaction-diffusion equations in a general form
Summary
Assume that we have an evolution continuous system, i.e., the family K of continuous maps φ: [0, +∞) ⟶ X:. Impulsive system {K, M, I} is an evolution system such that a phase point x(t) moves along trajectories of K until it reaches a fixed subset M ⊂ X which is called an impulsive set. At that moment τ (which is not fixed and depends on a particular trajectory), the phase point “jumps” into a new position x+(τ) ∈ Ix(τ), where I: M ⟶ X is called the impulsive map. Under suitable assumptions (see Definition 1), such an impulsive system can be considered as a multivalued (discontinuous) dynamical system. If this system is dissipative, its limit behavior can be described by the investigation of properties of uniform attractor, minimal compact uniformly attracting set in the phase space. We apply to study qualitative behavior of impulsive systems generated by reaction-diffusion evolution problems
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