Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems

In this article, we, first, consider a vibrating system of Timoshenko type in a one-dimensional bounded domain with an infinite history acting in the equation of the rotation angle. We establish a general decay of the solution for the case of equal-speed wave propagation as well as for the nonequal-speed case. We, also, discuss the well-posedness and smoothness of solutions using the semigroup theory. Then, we give applications to the coupled Timoshenko-heat systems (under Fourier's, Cattaneo's and Green and Naghdi's theories). To establish our results, we adopt the method introduced, in [13] with some necessary modifications imposed by the nature of our problems since they do not fall directly in the abstract frame of the problem treated in [13]. Our results allow a larger class of kernels than those considered in [28,29,30], and in some particular cases, our decay estimates improve the results of [28,29]. Our approach can be applied to many other systems with an infinite history.