Aissa Guesmia, Salim A. Messaoudi, Abdelaziz Soufyane
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 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 |
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Salim A. Messaoudi Mathematical Sciences Department KFUPM, Dhahran 31261, Saudi Arabia email: messaoud@kfupm.edu.sa |
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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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