Attractors for Second Order in Time Non-Conservative Dynamics with Nonlinear Damping

TL;DR

This paper investigates the long-term behavior of a nonlinear second-order in time PDE model for a suspension bridge, introducing a novel barrier's method to prove the existence of a finite-dimensional attractor despite non-dissipative effects.

Contribution

It develops a new barrier's method to establish the existence of a weak attractor and proves its strong, finite-dimensional nature in a complex nonlinear PDE system with non-conservative forces.

Findings

Existence of a weak attractor for the system.

Weak attractor is shown to be strong and finite-dimensional.

Novel barrier's method addresses challenges posed by non-dissipative effects.

Abstract

A long-time behavior of solutions to a nonlinear plate model subject to non-conservative and non-dissipative effects and nonlinear damping is considered. The model under study is a prototype for a suspension bridge under the effects of unstable flow of gas. To counteract the unwanted oscillations a damping mechanism of a nonlinear nature is applied. From the point of view of nonlinear PDEs, we are dealing with a non-dissipative and nonlinear second order in time dynamical system of hyperbolic nature subjected to nonlinear damping. One of the first goals is to establish ultimate dissipativity of all solutions, which will imply an existence of a weak attractor. The combined effects of non-dissipative forcing with nonlinear damping-leading to an overdamping-give rise to major challenges in proving an existence of an absorbing set. Known methods based on equipartition of the energy do not suffice. A rather general novel methodology based on ``barrier's'' method will be developed to address this and related problems. Ultimately, it will be shown that a weak attractor becomes strong, and the nonlinear PDE system has a coherent finite-dimensional asymptotic behavior.