On a Thermoelastic Laminated Timoshenko Beam: Well Posedness and Stability

Abstract

In this paper, we are concerned with a linear thermoelastic laminated Timoshenko beam, where the heat conduction is given by Cattaneo’s law. We firstly prove the global well posedness of the system. For stability results, we establish exponential and polynomial stabilities by introducing a stability number χ.

Exponential Stability of Laminated Beam With Constant Delay Feedback

“…For a system without structural damping, exponential and polynomial decay of the solution are possible in case of equal wave speeds, otherwise, the system lacks exponential stability. Other interesting results about damping through thermal effects can be found in [4,5,14] for thermoelasticity, and [19] for thermoelasticity of type III. In all these works, authors mainly established that the system decays exponentially in the case of equal wave speeds and polynomially otherwise, with and without structural damping.…”

Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term

“…1,2 Also, this is due to the relevance of the research topic to the applicability of these materials in the industry. Whereas, they presented a mathematical model of two-layer beams with structural damping due to the interslip obtained by 4 3 s + 4 3 s t − Ds xx = 0.…”

“…But if we add these conditions in all equations, the system decays exponentially without assuming equal wave speeds. [3][4][5][6] As for thermoelastic laminated Timoshenko beam, there are few results including the work of Liu et al 7 and Apalara. 8 In a previous study, 7 the authors considered the following laminated beams with past history: ( They established the global well posedness of solutions to the system.…”

“…If ≠ 0, they proved the exponential and polynomial stabilities depending on the behavior of the kernel function g and if = 0, they established exponential stability in case of equal wave speeds assumption, and exponential stability does not exist in case of nonequal wave speeds assumption. Apalara 8 considered a laminated beam with second sound of the following system: 4 3 s + 4 3 s t = 0, 3 t + q x + (3s − ) tx = 0, q t + q + x = 0 (1.3) and established the global well posedness and proved exponential and polynomial stabilities depending on the parameter .…”

“…Whereas, if we add linear damping terms to two of the three equations, System () is exponentially stable under the assumption (equal wave speeds) ( ρ / I ρ )=( G / D ). But if we add these conditions in all equations, the system decays exponentially without assuming equal wave speeds 3‐6 . As for thermoelastic laminated Timoshenko beam, there are few results including the work of Liu et al 7 and Apalara 8 .…”