Asymptotic behavior and optimality of the energy decay rate of certain Timoshenko systems

Résumé

In this thesis, we studied a nonlinear Timoshenko system with thermoelasticity with second sound. For this system we proved the strong stability using the Defermos’ technique which is based on Lasalle invariance principle . Then, we have used the comparison method in order to establish the lower estimate of the energy of the one-dimensional damped Timoshenko system with thermoelasticity . We have gone further and taken examples of nonlinear damping terms and we have shown that the lower bound of the energy yields the optimality of the decay rates for a specific example of the damping function g. Moreover, we have addressed an important problem in mathematical analysis of beam, namely the problem of determining the decay rate of the discrete energy by taking into account a few dissipative mechanisms. As we already know, Timoshenko system has two wave speeds and we have proved numerically that is sufficient to consider only one dissipation mechanism in order to obtain the exponential decay, for the case where the speeds are equal. Nevertheless, other dissipative cases have been considered in this thesis, we look for the behavior of the energy when the system is nonlinearly damped and we deduce an explicit (polynomial and logarithmic) decay rate of the discrete energy. One of the interesting feature of this work is the obtaining of the approximate values of the constants (coefficients and monomial degrees) of the decay rate function in time of the discrete energy associated with the Timoshenko system in an explicit manner.

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Statistiques

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