Exact decay rates for weakly coupled systems with indirect damping by fractional memory effects

This paper deals with the asymptotic behavior of a weakly coupled system of two equations in which one of them has a dissipative mechanism given by a memory term. This term depends on the fractional operator with exponent $\theta \in [0,1]$. We show that strong solutions of the system decay polynomially with a rate that depends on both the exponent θ and wave propagation speeds. Optimal decay rates are found and the results show a surprising aspect: More regular damping does not necessarily imply a faster decay.