Long-Time Behavior for a Thermoelastic Microbeam Problem with Time Delay and the Coleman-Gurtin Thermal Law

Liu, Wenjun; Chen, Dongqin; Chen, Zhijing

doi:10.1007/s10473-021-0220-3

Cited by 8 publications

(6 citation statements)

References 33 publications

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“…Other nonlinearities like p-Laplacian and fractional powers of operators are worked in [18][19][20][21][22]. Equations with delay terms are studied in [23,24]. Similar parabolic-hyperbolic systems to the ones presented in the introduction have been analyzed in [25,26].…”

Section: Preprintsmentioning

confidence: 99%

Global Non-existence of a Coupled Parabolic-Hyperbolic System of Thermoelastic Type with History

Esquivel-Avila

2023

Preprint

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We consider two abstract systems of parabolic-hyperbolic type that model thermoelastic problems. We study the influence of the physical constants and the initial data on the nonexistence of global solutions that in our framework are produced by the blow-up in finite time of the norm of the solution in the phase space. We employ a differential inequality to find sufficient conditions that produce the blow-up. To that end, we construct a set that is positive invariant for any positive value of the initial energy. As a result we found that the coupling with the parabolic equation stabilizes the system, as well as the damping term in the hyperbolic equation. Moreover, for any pair of positive values $(\xi,\epsilon)$, there exist initial data such that the corresponding solution with initial energy $\xi$ blows-up at a finite time less than $\epsilon$. Our purpose is to improve results previously published in the literature.

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Section: Preprintsmentioning

confidence: 99%

Global Non-existence of a Coupled Parabolic-Hyperbolic System of Thermoelastic Type with History

Esquivel-Avila

2023

Preprint

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“…There are also some papers on the control of the stability of coupled systems by the dissipation mechanism of heat. We can refer to previous studies [15][16][17][18] and the corresponding references. Other papers on the coupled systems with memory viscoelastic terms and Kelvin-Voigt damping have also been studied, which can be referred to previous studies [19][20][21][22][23] and the related references.…”

Section: Introductionmentioning

confidence: 99%

Stabilization for weakly coupled string–riser system with partial frictional damping

Zhang,

Liu

2023

Math Methods in App Sciences

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In this paper, we consider the stabilization of a 1‐D coupled string–riser system with partial frictional damping. The frictional damping is only distributed in the part of the string's or riser's domain, and its dissipative effect is transmitted to the other equation through the coupling. Firstly, we prove the existence and uniqueness result for solution of coupled system. Next, using a semigroup approach combined with the multipliers technique, we obtain that the two coupled systems are not exponentially stable, provide precise polynomial decay rates for their energy functionals, respectively, regardless of whether the friction damping only acts on the string equation or the riser equation, and give the optimality of polynomial decay rates. Finally, we compare the case of friction damping acting on string and riser equations simultaneously with the previous two cases. The results show that the case of friction damping acting on string and riser equations simultaneously is exponential stable, while the case of friction damping only acting on string or riser equations is polynomial stable and the decay result of friction damping only acts on the string equation is better than that only acts on the riser equation.

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“…Recently, the control of partial differential equations with time delay has become a hot research topic, see [2,5,18,20,21,22,28]. In [28], under the condition 0…”

Section: Introductionmentioning

confidence: 99%

Exponential stabilization for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights

Kong,

Liu,

2021

Preprint

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This paper is concerned with system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. By combining the semigroup theory with Kato's variable norm technique, we obtain the existence and uniqueness of solution. By imposing appropriate assumptions on the time-varying delay term and time-dependent weights, we introduce suitable perturbed Lyapunov functional to obtain exponential stability estimates.

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Long-Time Behavior for a Thermoelastic Microbeam Problem with Time Delay and the Coleman-Gurtin Thermal Law

Cited by 8 publications

References 33 publications

Global Non-existence of a Coupled Parabolic-Hyperbolic System of Thermoelastic Type with History

Global Non-existence of a Coupled Parabolic-Hyperbolic System of Thermoelastic Type with History

Stabilization for weakly coupled string–riser system with partial frictional damping

Exponential stabilization for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights

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