New decay results for a viscoelastic-type Timoshenko system with infinite memory

Abstract

This paper is concerned with the following memory-type Timoshenko system

$$\begin{aligned} {\left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-K(\varphi _x+\psi )_x =0,\\ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+\displaystyle \int \limits _0^{+\infty } g(s)\psi _{xx}(t-s){\mathrm{d}}s=0,\\ \end{array}\right. } \end{aligned}$$

with Dirichlet boundary conditions, where g is a positive nonincreasing function satisfying, for some nonnegative functions \(\xi \) and G,

$$\begin{aligned}g'(t)\le -\xi (t)G(g(t)),\qquad \forall t\ge 0.\end{aligned}$$

Under appropriate conditions on \(\xi \) and G, we establish some new decay results that generalize and improve many earlier results in the literature such as Mustafa (Math Methods Appl Sci 41(1): 192–204, 2018), Messaoudi et al. (J Integral Equ Appl 30(1): 117–145, 2018) and Guesmia (Math Model Anal 25(3): 351–373, 2020). We consider the equal speeds of propagation case, as well as the nonequal-speed case. Moreover, we delete some assumptions on the boundedness of initial data used in many earlier papers in the literature.

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Acknowledgements

We thank KFUPM and University of Sharjah for their continuous support. The authors also thank the referee for her/his very careful reading and valuable comments. This work was initiated and finished during the visits of the third author to KFUPM in December 2018–January 2019 and December 2019–January 2020. This work is funded by KFUPM under Project #SB181018.

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Correspondence to Adel M. Al-Mahdi.

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Al-Mahdi, A.M., Al-Gharabli, M.M., Guesmia, A. et al. New decay results for a viscoelastic-type Timoshenko system with infinite memory. Z. Angew. Math. Phys. 72, 22 (2021). https://doi.org/10.1007/s00033-020-01446-x

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Keywords

  • Timoshenko system
  • General decay
  • Infinite memory
  • Relaxation function
  • Viscoelasticity

Mathematics Subject Classification

  • 35B37
  • 35L55
  • 74D05
  • 93D15
  • 93D20