Decay property of Timoshenko system in thermoelasticity

Said‐Houari, Belkacem; Kasimov, Aslan R.

doi:10.1002/mma.1569

Cited by 45 publications

(24 citation statements)

References 41 publications

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“…They proved that heat dissipation alone is sufficient to stabilize the system in both cases and then concluded that the Timoshenko–Fourier and the Timoshenko–Cattaneo systems have the same decay rate, which depends on a certain stability number (which is a function of the parameters of the system) as identified previously by Santos et al in for Timoshenko system in a bounded domain. See also and for similar results. For more papers related to the second sound, we refer the reader to , and the references thereinIn this present work, we consider

({, \begin{matrix} ρ_{1} φ_{tt} - κ {(φ_{x} + ψ)}_{x} + σ θ_{x} = 0, in (0, 1) \times (0, \infty), \\ ρ_{2} ψ_{tt} - b ψ_{xx} + κ (φ_{x} + ψ) = 0, in (0, 1) \times (0, \infty), \\ ρ_{3} θ_{t} + q_{x} + σ φ_{xt} = 0, in (0, 1) \times (0, \infty), \\ τ q_{t} + q + γ θ_{x} = 0, in (0, 1) \times (0, \infty), \\ φ (x, 0) = φ_{0} (x), φ_{t} (x, 0) = φ_{1} (x), θ (x, 0) = θ_{0} (x), in (0, 1), \\ ψ (x, 0) = ψ_{0} (x), ψ_{t} (x <...> \end{matrix})

…”

Section: Introductionsupporting

confidence: 63%

On the decay rates of Timoshenko system with second sound

Apalara

Messaoudi

Keddi

2015

Math Methods in App Sciences

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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.

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({, \begin{matrix} ρ_{1} φ_{tt} - κ {(φ_{x} + ψ)}_{x} + σ θ_{x} = 0, in (0, 1) \times (0, \infty), \\ ρ_{2} ψ_{tt} - b ψ_{xx} + κ (φ_{x} + ψ) = 0, in (0, 1) \times (0, \infty), \\ ρ_{3} θ_{t} + q_{x} + σ φ_{xt} = 0, in (0, 1) \times (0, \infty), \\ τ q_{t} + q + γ θ_{x} = 0, in (0, 1) \times (0, \infty), \\ φ (x, 0) = φ_{0} (x), φ_{t} (x, 0) = φ_{1} (x), θ (x, 0) = θ_{0} (x), in (0, 1), \\ ψ (x, 0) = ψ_{0} (x), ψ_{t} (x <...> \end{matrix})

…”

Section: Introductionsupporting

confidence: 63%

On the decay rates of Timoshenko system with second sound

Apalara

Messaoudi

Keddi

2015

Math Methods in App Sciences

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“…We refer to [12,13,16] for frictional dissipation case, [5,14,15] for thermal dissipation case, and [1,2,8,9] for memory-type dissipation case.…”

Section: Introductionmentioning

confidence: 99%

Decay Property for the Timoshenko System With Fourier's Type Heat Conduction

Mori

Kawashima

LeFloch³

2014

J. Hyper. Differential Equations

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We study the Timoshenko system with Fourier's type heat conduction in the one-dimensional (whole) space. We observe that the dissipative structure of the system is of the regularity-loss type, which is somewhat different from that of the dissipative Timoshenko system studied earlier by Ide-Haramoto-Kawashima. Moreover, we establish optimal L 2 decay estimates for general solutions. The proof is based on detailed pointwise estimates of solutions in the Fourier space. Also, we introuce here a refinement of the energy method employed by Ide-Haramoto-Kawashima for the dissipative Timoshenko system, which leads us to an improvement on their energy method.

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“…Other studies on the dissipative Timoshenko system can be found in the literature. We refer to [20,21] for frictional dissipation case, [8,25,26] for thermal dissipation case, and [1,3,13,14] for memory-type dissipation case.…”

Section: Known Resultsmentioning

confidence: 99%

Global existence and optimal decay rates for the Timoshenko system: The case of equal wave speeds

Mori

Kawashima

2015

Journal of Differential Equations

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This work first gives the global existence and optimal decay rates of solutions to the classical Timoshenko system on the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems ([30]) can not be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and highfrequency decompositions. As a by-product, the usual decay estimate of L 1 (R)-L 2 (R) type is also shown.

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Decay property of Timoshenko system in thermoelasticity

Cited by 45 publications

References 41 publications

On the decay rates of Timoshenko system with second sound

On the decay rates of Timoshenko system with second sound

Decay Property for the Timoshenko System With Fourier's Type Heat Conduction

Global existence and optimal decay rates for the Timoshenko system: The case of equal wave speeds

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