Stabilization of laminated beam with structural damping and a heat conduction of Gurtin–Pipkin's law

Abstract

In the present work, we analyze the stability properties of a one dimensional thermoelastic laminated beam with structural damping, where the heat conduction is given by Gurtin–Pipkin's law. We show that the associated solution semigroup is exponentially stable if and only if the new introduced stability number χ is null. Furthermore, if $\chi \ne 0$ we prove the lack of exponential stability. The proofs are based on the energy method and Gearhart–Herbst–Prüss–Huang theorem.

Keywords:

• Laminated beams
• Gurtin–Pipkin thermal law
• exponential stability
• lack of exponential stability

2020 Mathematics Subject Classifications:

• 35B35
• 35B40
• 74F05
• 93D20

Acknowledgments

The author thanks very much the anonymous referee for his/her respectful advice and remarks to correct and improve the paper.

Disclosure statement

No potential conflict of interest was reported by the author.