2024/01/12 by Bortolan, Matheus C., Caraballo, Tomas, Neto, Carlos Pecorari
#35B41 #35L20 #37L25 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.06631
In this work we introduce the concept of generalized exponential \mathfrakD-pullback attractor for evolution processes, where \mathfrakD is a universe of families in X, which is a compact and positively invariant family that pullback attracts all elements of \mathfrakD 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 \mathfrakDC^∗-pullback attractor for an evolution process, where \mathfrakDC^∗ 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 \mathfrakDC^∗-pullback attractor. This, in turn, also implies the existence of the \mathfrakDC^∗-pullback attractor for such problem.