Generalized exponential \({\mathfrak {D}}_{\mathcal {C}^*}\)–pullback attractor for a nonautonomous wave equation

Abstract

In this work we introduce the concept of generalized exponential \({\mathfrak {D}}_{\mathcal {C}^*}\)–pullback attractors for evolution processes, which are compact and positively invariant families that pullback attract all elements of a universe of families \({\mathfrak {D}}_{\mathcal {C}^*}\), with an exponential rate. Such concept, within the pullback framework for nonautonomous problems, was introduced in Bortolan et al. (Appl Math Optim 89(62):1–52, 2024) for more general decay functions (which include the exponential decay), but for fixed bounded sets rather than for a universe of families, and was inspired by Zhao et al. (Estimate of the attractive velocity of attractors for some dynamical systems, http://arxiv.org/abs/2108.07410, 2021), which dealt with the autonomous case. We prove a result that ensures the existence of a generalized exponential \({\mathfrak {D}}_{\mathcal {C}^*}\)–pullback attractor for an evolution process, using the concept of pullback \(\kappa \)–dissipativity for evolution processes with respect to a general universe \({\mathfrak {D}}\). This required an adaptation of the results presented in Bortolan et al. (Appl Math Optim 89(62):1–52, 2024), which only covered the case of a polynomial rate of attraction for fixed bounded sets. Later, we prove that a nonautonomous wave equation has a generalized exponential \({\mathfrak {D}}_{\mathcal {C}^*}\)–pullback attractor. This, in turn, also implies the existence of the \({\mathfrak {D}}_{\mathcal {C}^*}\)–pullback attractor for such problem.