# American Institute of Mathematical Sciences

## Dynamics of Timoshenko system with time-varying weight and time-varying delay

 1 Department of Mathematics, Federal University of Bahia, Salvador, 40170-115, Bahia, Brazil 2 Faculty of Exact Sciences and Technology, Federal University of Pará, Manoel de Abreu Street, s/n, 68440-000, Abaetetuba, Pará, Brazil 3 Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, 36307-352, Minas Gerais, Brazil

* Corresponding author: raposo@ufsj.edu.br

Received  May 2020 Revised  November 2020 Published  February 2021

Fund Project: The first author was partially supported by CAPES (Brazil)

This paper is concerned with the well-posedness of global solution and exponential stability to the Timoshenko system subject with time-varying weights and time-varying delay. We consider two problems: full and partially damped systems. We prove existence of global solution for both problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the Energy Method. For partially damped system the exponential stability is proved under assumption of equal-speed wave propagation in the transversal and angular directions. For full damped system the exponential stability is obtained without the hypothesis of equal-speed wave propagation.

Citation: Carlos Nonato, Manoel Jeremias dos Santos, Carlos Raposo. Dynamics of Timoshenko system with time-varying weight and time-varying delay. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021053
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