Electron. J. Differential Equations, Vol. 2022 (2022), No. 02, pp. 1-26.

Decay rates for two Cauchy thermoelastic laminated Timoshenko problems of type III with interfacial slip

Aissa Guesmia

In this article we study the decay of solutions for two systems of laminated Timoshenko beams with interfacial slip, in the whole space R subject to a thermal effect of type III acting only on one component. When the thermal effect acts via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we prove that both systems are polynomially stable. Also we obtain stability estimates in the L2(R)-norm of solutions and their higher order derivatives with respect of the space variable. The decay rates, and the absence or presence of the regularity-loss type property, depend on the regularity of the initial data and the speeds of wave propagations. However, when the thermal effect acts via the first component (transversal displacement), we introduce a new stability number χ and prove that the stability of the system is equivalent to χ≠0. An application to a case of lower order coupling terms is also given. To prove our results, we use the energy method in the Fourier space combined with well chosen weight functions to build appropriate Lyapunov functionals.

Submitted January 9, 2021. Published January 5, 2022.
Math Subject Classifications: 34B05, 34D05, 34H05.
Key Words: Timoshenko beam; interfacial slip; heat conduction; energy method; Fourier analysis.

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Aissa Guesmia
Institut Elie Cartan de Lorraine
UMR 7502, Université de Lorraine
3 Rue Augustin Fresnel, BP 45112
57073 Metz Cedex 03, France
email: aissa.guesmia@univ-lorraine.fr

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