Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 193, pp. 1-45.

Title: Stabilization of a linear Timoshenko system with infinite history 
       and applications to the Timoshenko-heat systems
 
Authors: Aissa Guesmia  (Metz Univ., France)
         Salim A. Messaoudi (KFUPM, Dhahran, Saudi Arabia)
	 Abdelaziz Soufyane (Alhosn Univ., Abu Dhabi, UAE)
	 
Abstract:
 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.

Submitted April 9, 2012. Published November 06, 2012.
Math Subject Classifications: 35B37, 35L55, 74D05, 93D15, 93D20.
Key Words: General decay; infinite history; relaxation function; 
           Timoshenko; thermoelasticity.