On the stability of a laminated beam with structural damping and Gurt–Pipkin thermal law

Abstract

In this paper, we investigate the stabilization of a one-dimensional thermoelastic laminated beam with structural damping coupled with a heat equation modeling an expectedly dissipative effect through heat conduction governed by Gurtin–Pipkin thermal law. Under some assumptions on the relaxation function g, we establish the well-posedness of the problem by using Lumer–Phillips theorem. Furthermore, we prove the exponential stability and lack of exponential stability depending on a stability number by using the perturbed energy method and Gearhart–Herbst–Prüss–Huang theorem, respectively.

Keywords

• laminated beam
• Gurtin–Pipkin thermal law
• well-posedness
• exponential stability
• lack of exponential stability