Electron. J. Diff. Equ., Vol. 2012 (2012), No. 193, pp. 1-45.

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

Aissa Guesmia, Salim A. Messaoudi, Abdelaziz Soufyane

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 6, 2012.
Math Subject Classifications: 35B37, 35L55, 74D05, 93D15, 93D20.
Key Words: General decay; infinite history; relaxation function; Timoshenko; thermoelasticity.

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Aissa Guesmia
Laboratory of Mathematics and Applications of Metz
Bat. A, Lorraine - Metz University
Ile de Sauley, 57045 Metz Cedex 01, France
email: guesmia@univ-metz.fr
Salim A. Messaoudi
Mathematical Sciences Department
KFUPM, Dhahran 31261, Saudi Arabia
email: messaoud@kfupm.edu.sa
Abdelaziz Soufyane
College of Engineering and Applied Sciences
Alhosn University, P.O. Box 38772, Abu Dhabi, UAE
email: a.soufyane@alhosnu.ae

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