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  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 . 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
| Salim A. Messaoudi |
Mathematical Sciences Department
KFUPM, Dhahran 31261, Saudi Arabia
| Abdelaziz Soufyane |
College of Engineering and Applied Sciences
Alhosn University, P.O. Box 38772, Abu Dhabi, UAE
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