New stability result of a partially dissipative viscoelastic Timoshenko system with a wide class of relaxation function

Abstract

This paper is concerned with the following partially dissipative viscoelastic Timoshenko system $\{\begin{array}{l}\begin{array}{l}{\rho }_1{\varphi }_{tt}-\kappa ({\varphi }_x+\psi {)}_x\\ +\kappa {\int }_0^{t}g(t-s)({\varphi }_x+\psi {)}_x\mathrm{d}s=0& \mathrm{in}\left(0,L\right)\times {\mathbb{R}}^{+},\\ {\rho }_2{\psi }_{tt}-b{\psi }_{xx}+\kappa ({\phi }_x\\ +\psi )-\kappa {\int }_0^{t}g(t-s)({\varphi }_x+\psi )\mathrm{d}s=0& \mathrm{in}\left(0,L\right)\times {\mathbb{R}}^{+},\end{array}\end{array}$with damping mechanism acting only on the shear force, and with Dirichlet boundary conditions. We consider a very general relaxation function $g^{\prime }(t)\le -\xi (t)H(g(t)),\mathrm{\forall }t\ge 0.$Under appropriate conditions on ξ and H, we establish a general stability result. The result is obtained under the assumption of equal speed of wave propagation. This work extends and generalizes many results in literature such as Alves et al. [On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J Math Anal. 2019;51(6):4520–4543].

Keywords:

• Timoshenko system
• viscoelasticity
• general decay
• convex functions
• equal wave speeds

2020 Mathematics Subject Classifications:

• 35L53
• 35B35
• 35B40
• 74D05
• 93D20

Acknowledgments

The authors would like to express their sincere thanks to King Fahd University of Petroleum and Minerals for its continuous support. Additionally, the author would like to thank the reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is funded by King Fahd University of Petroleum and Minerals (KFUPM) – Interdisciplinary Research Center (IRC) for Construction and Building Materials. This work is supported by KFUPM under project #SR181024.