Stability of Timoshenko systems with past history

In this work, we study the well‐posedness and the asymptotic stability of a one‐dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number χτ=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd.

“…Whereas, in the opposite case

\frac{κ}{ρ_{1}} \neq \frac{b}{ρ_{2}}

Section: Introductionsupporting

confidence: 80%

On the decay rates of Timoshenko system with second sound

Apalara

Messaoudi

Keddi

2015

“…With respect to viscoelastic damping, Ammar Khodja et al . considered the memory effect, and Muñoz Rivera and Fernández Sare treated the Timoshenko systems with a past history under suitable conditions on the relaxation functions. This paper has been improved by Messauodi and Said‐Houari in by using conditions on the relaxation function weaker than those in .…”

Section: Introductionmentioning

confidence: 99%

Stability to weakly dissipative Timoshenko systems

Júnior

Santos

Rivera

2013

“…The same problem with an additional damping of history type of the form

{MathClass-op\int}_{0}^{MathClass-rel\infty} g (s) ψ_{xx} (x MathClass-punc, t MathClass-bin- s) normald s

acting in the second equation has been analyzed in . The authors of proved exponential and polynomial stability results for equal as well as non‐equal wave speeds under conditions on the relaxation function g weaker than those in .…”

Section: Introductionmentioning

confidence: 99%

Decay property of Timoshenko system in thermoelasticity

Said‐Houari

Kasimov

2011

We investigate the decay property of a Timoshenko system of thermoelasticity in the whole space for both Fourier and Cattaneo laws of heat conduction. We point out that although the paradox of infinite propagation speed inherent in the Fourier law is removed by changing to the Cattaneo law, the latter always leads to a solution with the decay property of the regularity‐loss type. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We derive L2 decay estimates of solutions and observe that for the Fourier law the decay structure of solutions is of the regularity‐loss type if the wave speeds of the first and the second equations in the system are different. For the Cattaneo law, decay property of the regularity‐loss type occurs no matter what the wave speeds are. In addition, by restricting the initial data to U0MathClass-rel∈Hs(double-struckR)MathClass-bin∩L1MathClass-punc,γ(double-struckR) with a suitably large s and γ ∈ [0,1], we can derive faster decay estimates with the decay rate improvement by a factor of t−γ/2. Copyright © 2011 John Wiley & Sons, Ltd.