Global Attractors for an Extensible Thermoelastic Beam System

TL;DR

This paper investigates the long-term behavior of an extensible thermoelastic beam system, proving the existence of a global attractor under various conditions, which describes the system's stable states over time.

Contribution

It establishes the existence of a global attractor with optimal regularity for a thermoelastic beam system, including cases with rotational inertia and time-independent forces.

Findings

Existence of bounded absorbing sets

Presence of a global attractor with optimal regularity

Solutions on the attractor are strong solutions

Abstract

This work is focused on the dissipative system describing the dynamics of an extensible thermoelastic beam, where the dissipation is entirely contributed by the second equation ruling the evolution of the temperature. Under natural boundary conditions, we prove the existence of bounded absorbing sets. When both the external body force and the heat source are time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity for all values of the external axial load. The same result holds true when the rotational inertia is taken into consideration. In both cases, the solutions on the attractor are strong solutions.