Sobre sistemas dissipativos com amortecimento do tipo derivada fracionária: dos semigrupos aos processos evolutivos.

Abstract

This work addresses the analysis of three evolution problems with fractional derivative-type damping, investigating the existence, uniqueness, and asymptotic behavior of solutions. The first problem consists of a one-dimensional linear and autonomous model of a suspension bridge, whose deck is modeled by Timoshenko Beam Theory. The system incorporates fractional damping terms in each of its equations. For this model, the Theory of Semigroups of Bounded Linear Operators was applied to demonstrate the existence and uniqueness of global solution. The asymptotic analysis revealed that the energy decay of the system is not exponential but rather polynomial. The second problem addresses an abstract, nonlinear, autonomous N-dimensional model for a suspension bridge, governed by Kirchhoff plate theory for the deck and again subject to fractional damping. The proof of local solution existence was achieved using Classical Semigroup Theory. The demonstration that this solution is global (i.e., does not blow up in finite time) was carried out via energy estimates for the solution norms. The long-term behavior analysis was conducted using the Theory of Nonlinear Semigroups of continuous operators (dynamical systems), which established the existence of a compact global attractor that attracts all system trajectories. Finally, the third problem analyzes a nonlinear and non-autonomous wave equation model with an acoustic boundary condition, subject to a nonlinear internal damping and a fractional derivative-type damping on the boundary. The existence of a local solution was established by combining Semigroup Theory with Kato’s Cauchy-Duhamel (CD) Systems Theory. The proof that these solutions are global again followed from energy estimates. For the asymptotic study, the Theory of Evolutionary Processes, which generalizes the notion of semigroups to the non-autonomous context, was used. Through this theory, it was demonstrated that the solutions admit a time-dependent family of compact sets (a pullback attractor) that attracts the trajectories in the pullback sense, i.e., when solutions evolve from initial conditions taken at times increasingly remote in the past.