Stability of a uniform attractor for a second-order evolution equation with discontinuous trajectories

Abstract

The work is devoted to the study of the qualitative behavior of solutions of the wave equation, whose trajectories have impulsive perturbations in moments when they reach a fixed (impulsive) subset of the phase space. Using the general constructing scheme of the infinite-dimensional impulsive dynamical system and using the concept of a uniform attractor — a minimal compact uniformly attracting set, we obtained the result about the existence and the explicit form of a uniform attractor for the corresponding impulsive dynamical system. Trajectories of such system can have an infinite number of impulsive points when they encounter an impulsive subset of the phase space. Thus, a uniform attractor may have a non-empty intersection with the impulsive set and, as a result, may not have the stability property. However, due to the additional conditions on the impulsive parameters of the problem, we managed to prove the stability property for the non-impulsive part of the uniform attractor.