Indirect stabilization of weakly coupled Kirchhoff plate and wave equations with frictional damping

Hajej, Ahmed; Hajjej, Zayd; Tebou, Louis

doi:10.1016/j.jmaa.2019.01.046

Cited by 13 publications

(6 citation statements)

References 23 publications

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“…Before proving the above theorem, we want to compare the polynomial estimate obtained here with the one established in [18,Theorem 3.1].…”

Section: Introductionmentioning

confidence: 99%

“…Such a system is weakly coupled and falls within the general framework of the indirect stabilization of weakly coupled elastic systems, which has quite a rich literature, e.g. [1,2,3,16,18,35,41,43,45]. Unlike the works just cited dealing with the indirect stabilization of weakly coupled elastic systems, where one system is mechanically damped and the other one undamped, we are dealing here with a different type of indirect stabilization problem; more precisely, we are dealing with a doubly indirect stabilization problem in the sense that we are relying on the dissipation induced by the heat component of the system to strongly stabilize the coupled system.…”

Section: Introductionmentioning

confidence: 99%

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Stabilization of coupled thermoelastic Kirchhoff plate and wave equations

Mansouri,

Tebou

2020

ejde

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We consider a coupled system consisting of a Kirchhoff thermoelastic plate and an undamped wave equation. It is known that the Kirchhoff thermoelastic plate is exponentially stable. The coupling is weak. First, we show that the coupled system is not exponentially stable. Afterwards, we prove that the coupled system is polynomially stable, and provide an explicit polynomial decay rate of the associated semigroup. Our proof relies on a combination of the frequency domain method and the multipliers technique. For more information see https://ejde.math.txstate.edu/Volumes/2020/121/abstr.html

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“…Before proving the above theorem, we want to compare the polynomial estimate obtained here with the one established in [18,Theorem 3.1].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stabilization of coupled thermoelastic Kirchhoff plate and wave equations

Mansouri,

Tebou

2020

ejde

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“…Zhu [4] devoted to the study of stability of weakly coupled plate equations with local distributed damping, and it is proved that the energy of the correlated weakly dissipative coupled systems decreases logarithmically. For the coupled system of Kirchhoff plate/wave and Euler–Bernoulli plate/wave in Ahmed et al [5] and Tebou [6], they proved the polynomial decay results and obtained that the damping additive on the wave is better than the damping additive on the Kirchhoff plate or Euler–Bernoulli plate. For more literatures on polynomial decay results, we can also refer to previous studies [7–10].…”

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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“…For indirect stabilization, Alabau et al [23] considered the stabilization of an abstract system of two coupled second-order evolution equations, wherein only one of the equations is stabilized and showed that the energy decays polynomially. Recently, Hajej et al [24] studied the indirect stabilization (only one equation of the coupled system is damped) of a coupled wave equation and Kirchhoff plate equation without viscoelastic terms (g 1 = g 2 = 0), and with frictional damping, the polynomial decay was derived. Motivated by these works, in this paper, we study the stability of this coupled system but only with the presence of viscoelastic terms in the two equations with a wider class of relaxation functions.…”

Section: Introductionmentioning

confidence: 99%

Existence and General Decay of Solutions for a Weakly Coupled System of Viscoelastic Kirchhoff Plate and Wave Equations

Hajjej

2023

Symmetry

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In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions.

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Indirect stabilization of weakly coupled Kirchhoff plate and wave equations with frictional damping

Cited by 13 publications

References 23 publications

Stabilization of coupled thermoelastic Kirchhoff plate and wave equations

Stabilization of coupled thermoelastic Kirchhoff plate and wave equations

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

Existence and General Decay of Solutions for a Weakly Coupled System of Viscoelastic Kirchhoff Plate and Wave Equations

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