Quasi-stability and attractors for a nonlinear coupled wave system with memory

Santos, M. J. Dos; Lobato, R. F. C.; Cordeiro, Sebastião Martins Siqueira; Santos, Ana Carlota B. dos

doi:10.1007/s40574-020-00258-1

Cited by 7 publications

(6 citation statements)

References 12 publications

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“…The following two results display the strong properties of the quasi‐stability for a dynamical system (see Chapter 7 in Chueshov and Lasiecka [1] or previous works [8, 35, 36]).…”

Section: Finite Dimensional Global Attractor For the System With μ2=...mentioning

confidence: 54%

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Well‐posedness and dynamics for the von Karman equation with memory and nonlinear time‐varying delay

Cui

2023

Math Methods in App Sciences

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This paper is concerned with mathematical analysis for the viscoelastic von Karman equation with memory and time‐varying delay. Based on the existence and uniqueness of the strong solution established by Galerkin method, the finite dimensional global attractor for the system without time‐varying delay has been shown by using the quasi‐stability theory. The difficulties for our desired well‐posedness are the limiting of fourth‐order derivative terms and the von Karman bracket term, which need some delicate estimates to achieve. The key point for the dynamics is to obtain the quasi‐stability for gradient system, which needs the higher regularity of trajectory especially the von Karman bracket.

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“…The following two results display the strong properties of the quasi‐stability for a dynamical system (see Chapter 7 in Chueshov and Lasiecka [1] or previous works [8, 35, 36]).…”

Section: Finite Dimensional Global Attractor For the System With μ2=...mentioning

confidence: 54%

“…The existence of the attracting compact set and the asymptotically compactness or the asymptotically smoothness of the dynamical system assure the existence of the global attractor as in the following theorem (see earlier studies [8, 35–39]).…”

Section: Finite Dimensional Global Attractor For the System With μ2=...mentioning

confidence: 99%

Well‐posedness and dynamics for the von Karman equation with memory and nonlinear time‐varying delay

Cui

2023

Math Methods in App Sciences

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“…The authors showed that the system is strongly controllable in the energy space for a dense set of system parameters with V (t) = p t ( , t). Ramos et al in [23] inserted the damping term δv t with δ > 0 in the equation (3) and considered V (t) = 0 in (6). The authors obtained an exponential stability result for the system (3)-( 6) as well as for its discrete counterpart.…”

mentioning

confidence: 99%

“…In [24] Ramos et al inserted in the equation (3) the term damping ξ 1 v t (x, t) and damping with delay ξ 2 v t (x, t − τ ), where ξ 1 , ξ 3 and τ > 0 are constant, and V (t) = 0 in (6). The authors obtained exponential stability for the model, they consider the relationship ξ 2 < ξ 1 .…”

mentioning

confidence: 99%

“…In this work, we propose to analyze the issue of exponential stability for the system (3)-( 6) (which is a system of two coupled wave equations) with past history inserted in the longitudinal displacement equation (3) with V (t) = 0 in (6), that is, we consider the following system (9) µp tt − βp xx + γβv xx = 0, in (0, ) × (0, ∞), (10) with boundary conditions v(0, t) = p(0, t) = v x ( , t) = p x ( , t) = 0, t > 0, (11) and initial data…”

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confidence: 99%

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Exponential stability for a piezoelectric beam with a magnetic effect and past history

Santos

Fortes

Cardoso

2022

DCDS-B

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<p style='text-indent:20px;'>Solutions for systems consisting of coupled wave equations, one of them with past history, may present different behaviors due to the type of coupling. In this paper, the issue of exponential stability for a piezoelectric beam with magnetic effect and past history is analyzed. In the work is proved that the past history term acting on the longitudinal motion equation is sufficient to cause the exponential decay of the semigroup associated with the system, independent of any relation involving the model coefficients.</p>

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On the energy decay of a viscoelastic piezoelectric beam model with nonlinear internal forcing terms and a nonlinear feedback

Al‐Gharabli,

Messaoudi

2023

Math Methods in App Sciences

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In this paper, we consider a piezoelectric beam model with nonlinear source terms and investigate the interaction between a viscoelastic damping and a nonlinear frictional damping. Under general assumptions on the relaxation function and the nonlinear feedback, we establish explicit formulae for the energy decay rates of this system and prove that the energy decays at a rate dictated by the weaker damping. Our results substantially improve some earlier related results in the literature.

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Quasi-stability and attractors for a nonlinear coupled wave system with memory

Cited by 7 publications

References 12 publications

Well‐posedness and dynamics for the von Karman equation with memory and nonlinear time‐varying delay

Well‐posedness and dynamics for the von Karman equation with memory and nonlinear time‐varying delay

Exponential stability for a piezoelectric beam with a magnetic effect and past history

On the energy decay of a viscoelastic piezoelectric beam model with nonlinear internal forcing terms and a nonlinear feedback

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