We study the asymptotic behavior of Timoshenko systems with memory, where the memory is given by a non-dissipative kernel and is acting only on one equation of the system. We show that the exponential stability depends on conditions regarding the decay rate of the kernel and a nice relationship between the coefficients of the system. Moreover, with full-memory effect in the system, we will show exponential stability in the general case.
"Exponential decay of Timoshenko systems with indefinite memory dissipation." Adv. Differential Equations 13 (7-8) 733 - 752, 2008.