Attracting sets in sup-norms for mild solutions of impulsive-perturbed parabolic semilinear problems

In this paper, we investigate the qualitative behavior of an evolutionary problem that consists of a semilinear parabolic equation whose trajectories undergo instantaneous impulsive perturbations at the moments when some integral functional reaches a certain threshold value. The key object is the uniform attractor of the corresponding impulsive infinite-dimensional dynamical system. The novelty of this study is the analysis of mild solutions in the phase space of continuous functions. Under general assumptions on the impulsive parameters, we prove that this problem generates an impulsive dynamical system, and its trajectories have a compact uniform attractor with respect to the supremum norm (sup-norm).