Qualitative properties of solutions to a nonlinear transmission problem for an elastic Bresse beam

TL;DR

This paper analyzes a nonlinear transmission problem for a Bresse beam with damping, proving well-posedness, existence of a global attractor, and studying singular limits with numerical modeling.

Contribution

It establishes well-posedness and global attractor existence for a nonlinear Bresse beam problem with damping, and investigates singular limits through numerical simulations.

Findings

Well-posedness of the PDE system in energy space

Existence of a regular global attractor under certain conditions

Numerical modeling of singular limits as parameters vary

Abstract

We consider a nonlinear transmission problem for a Bresse beam, which consists of two parts, damped and undamped. The mechanical damping in the damped part is present in the shear angle equation only, and the damped part may be of arbitrary positive length. We prove well-posedness of the corresponding PDE system in energy space and establish existence of a regular global attractor under certain conditions on nonlinearities and coefficients of the damped part only. Moreover, we study singular limits of the problem when $l\to 0$ or $l\to 0$ simultaneously with $k_i\to +\infty$ and perform numerical modelling for these processes.