Some well-posedness and general stability results in Timoshenko systems with infinite memory and distributed time delay

In this paper, we consider a Timoshenko system in one-dimensional bounded domain with infinite memory and distributed time delay both acting on the equation of the rotation angle. Without any restriction on the speeds of wave propagation and under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove, first, the well-posedness and, second, the stability of the system, where we present some decay estimates depending on the equal-speed propagation case and the opposite one. The obtained decay rates depend on the growths of the memory and delay kernels at infinity. In the nonequal-speed case, the decay rate depends also on the regularity of initial data. Our stability results show that the only dissipation resulting from the infinite memory guarantees the asymptotic stability of the system regardless to the speeds of wave propagation and in spite of the presence of a distributed time delay. Applications of our approach to specific coupled