Stability Results for a Timoshenko System with a Fractional Operator in the Memory

Abstract

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.