Stability for a coupled system of second-order evolution equations with indirect memory-damping

We study the stability for a coupled system of second-order evolution equations with indirect memory-damping, which arises from the theory of viscoelasticity and takes the Timoshenko systems as concrete examples. What concerns us is how to efficiently control the memory kernels to achieve the desired stabilities of the whole system. With the help of the memory and coupling effects in the system, we establish some optimal stability results for the energy of the system under weak conditions. Especially, we do not need the usual nonnegative/nonincreasing restrictions on the memory kernels in our results. For the Timoshenko systems, we obtain the optimal decay rate under minimal assumptions on memory kernels and the best (so far) exponential stability result in the case of equal wave speeds. As can be seen, to overcome the difficulties in obtaining these results, novel ideas and techniques are presented and utilized in our proofs.