Generalized exponential pullback attractor for a nonautonomous wave equation

Abstract

In this work we introduce the concept of generalized exponential $\mathfrak{D}$-pullback attractor for evolution processes, where $\mathfrak{D}$ is a universe of families in $X$, which is a compact and positively invariant family that pullback attracts all elements of $\mathfrak{D}$ with an exponential rate. Such concept was introduced in arXiv:2311.15630 for the general case of decaying functions (which include the exponential decay), but for fixed bounded sets rather than to universe of families. We prove a result that ensures the existence of a generalized exponential $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for an evolution process, where $\mathfrak{D}_{\mathcal{C}^\ast}$ is a specific universe. This required an adaptation of the results of arXiv:2311.15630, 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 $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor. This, in turn, also implies the existence of the $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for such problem.