On the asymptotic properties of solutions to a nonlinear transmission problem for a Bresse beam with thermal damping

TL;DR

This paper investigates the long-term behavior of solutions to a nonlinear transmission problem in a Bresse beam with thermal damping, proving well-posedness, existence of attractors, and analyzing singular limits with numerical simulations.

Contribution

It establishes well-posedness and the existence of a global attractor for a nonlinear Bresse beam system with thermal damping, and explores singular limits with numerical modeling.

Findings

Proved well-posedness of the nonlinear system.

Established existence of a compact global attractor.

Analyzed singular limits as curvature tends to zero and shear moduli tend to infinity.

Abstract

A nonlinear transmission problem for an arch beam, which consists of two parts with different material properties is considered. One of the parts is subjected to thermal damping while another one is undamped. The thermal damping affects only the shear angle and the longitudinal equation within the damped part, which can have any positive length. We prove the well-posedness of the system in the energy space and establish the existence of a compact global attractor under certain conditions imposed solely on the coefficients of the damped part. Additionally, we study singular limits of the system as the curvature of the beam tends to zero or in case both the curvature vanishes and the shear moduli tend to infinity. Numerical simulations are also carried out to model these limiting behaviors.