Existence and Invariance of Global Attractors for Impulsive Parabolic System Without Uniqueness

Abstract

In this paper, we apply the abstract theory of global attractors for multivalued impulsive dynamical systems to weakly-nonlinear impulsively perturbed parabolic system without uniqueness of a solution to the Cauchy problem. We prove that for a sufficiently wide class of impulsive perturbations (including multi-valued ones) the global attractor of the corresponding multi-valued impulsive dynamical system has an invariant non-impulsive part.

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

“…The systematic study of these systems began relatively recently and primarily focused on systems defined in the finite-dimensional phase spaces, e.g., on systems defined in Euclidean space R n , n ∈ N [30][31][32][33][34] and the so-called multi-frequency systems defined in the product of a torus and Euclidean space T m × R n , n, m ∈ N [35][36][37][38]. The results regarding the limit behavior of infinite-dimensional impulsive dynamical systems can be found in [39][40][41][42][43][44][45]. However, in both the parabolic and hyperbolic cases, the impulsive parameters are "finite-dimensional" in nature, i.e., only a finite number of coordinates of the phase vector is subjected to an impulsive disturbance.…”

“…Proof of Lemma 1. According to [40], the nonempty ω-limit set of G(t, z 0 ) for any initial condition z 0 ∈ X exists if the sequence…”

“…It is important to note that we cannot expect ω(z 0 ) to be stable in any sense since this is not true even in the non-impulsive case. The stability property can be guaranteed for more massive objects, namely for uniform attractors [40]. Nevertheless, we can ensure the invariance of the non-impulsive part of ω(z 0 ).…”

“…Proof of Lemma 2. We use the following fact, proven in a much more general situation in [40]: Under the conditions of ( 18), if…”

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

“…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.…”