Stability Results for a Timoshenko System with a Fractional Operator in the Memory
- 1 Downloads
We study the asymptotic behavior of Timoshenko systems with a fractional operator in the memory term depending on a parameter \(\theta \in [0,1]\) and acting only on one equation of the system. Considering exponentially decreasing kernels, we find exact decay rates. To be precise, we show that for \(\theta \in [0,1)\), the system decay polynomially with rates that depend on the value of the difference of the wave propagation speeds. We also prove that these decay rates are optimal. Moreover, when \(\theta =1\) and the equations have the same propagation speeds we obtain the exponential decay of the solutions.
KeywordsTimoshenko system Fractional damping Memory term Exponential stability Polynomial decay
The first author has been partially supported by PNPD/CAPES (CAPES Postdoctoral National Program).
Compliance with Ethical Standards
Conflict of interest
The authors declare that they have no conflict of interest.
- 2.Almeida Júnior, D.S., Ramos, A.J.A.: On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency. Z. Angew. Math. Phys. 68(6), Article 145 (2017)Google Scholar
- 10.Benaissa, A., Benazzouz, S.: Well-posedness and asymptotic behavior of Timoshenko beam system with dynamic boundary dissipative feedback of fractional derivative type. Z. Angew. Math. Phys. 68, Article 94 (2017)Google Scholar
- 21.Oquendo, H.P., Raya, R.P.: Best rates of decay for coupled waves with different propagation speeds. Z. Angew. Math. Phys. 68, Art. 77 (2017)Google Scholar
- 30.Tian, X., Zhang, Q.: Stability of a Timoshenko system with local Kelvin–Voigt damping, Z. Angew. Math. Phys. 68, Article 20 (2017)Google Scholar