Asymptotic behavior to Bresse system with past history

Santos, Mauro de Lima; Soufyane, Abdelaziz; Júnior, D. S. Almeida

doi:10.1090/s0033-569x-2014-01382-4

Cited by 37 publications

(41 citation statements)

References 17 publications

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“…Similar stability results to the ones of [11] were also proved in [9] under one infinite memory acting on the longitudinal displacement (that is g 1 = g 2 = 0) with kernels having a general decay at infinity, and in [6] under one infinite memory acting on the shear angle displacement (that is g 1 = g 3 = 0) with kernels having an esponential decay at infinity.…”

Section: Introductionsupporting

confidence: 75%

“…On the other hand, taking the inner product of (4.20) 6 with wn λ 4 n in L 2 (0, 1), using (4.18), integrating by parts and using the boundary conditions, we observe that …”

Section: )mentioning

confidence: 99%

“…Taking the inner product of (4.20) 4 with wn λ 4 n and of (4.20) 6 with ψn λ 4 n in L 2 (0, 1), and using (4.18), we…”

Section: )mentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement

Guesmia

2017

Nonautonomous Dynamical Systems

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Abstract:The asymptotic stability of one-dimensional linear Bresse systems under infinite memories was obtained by Guesmia and Kafini [10] (three infinite memories), Guesmia and Kirane [11] (two infinite memories), Guesmia [9] (one infinite memory acting on the longitudinal displacement) and De Lima Santos et al. [6] (one infinite memory acting on the shear angle displacement). When the kernel functions have an exponential decay at infinity, the obtained stability estimates in these papers lead to the exponential stability of the system if the speeds of wave propagations are the same, and to the polynomial one with decay rate t −1 2 otherwise. The subject of this paper is to study the case where only one infinite memory is considered and it is acting on the vertical displacement. As far as we know, this case has never studied before in the literature. We show that this case is deeply different from the previous ones cited above by proving that the exponential stability does not hold even if the speeds of wave propagations are the same and the kernel function has an exponential decay at infinity. Moreover, we prove that the system is still stable at least polynomially where the decay rate depends on the smoothness of the initial data. For classical solutions, this decay rate is arbitrarily close to t −1 4 . The proof is based on a combination of the energy method and the frequency domain approach to overcome the new mathematical difficulties generated by our system.

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Section: Introductionsupporting

confidence: 75%

“…On the other hand, taking the inner product of (4.20) 6 with wn λ 4 n in L 2 (0, 1), using (4.18), integrating by parts and using the boundary conditions, we observe that …”

Section: )mentioning

confidence: 99%

“…Taking the inner product of (4.20) 4 with wn λ 4 n and of (4.20) 6 with ψn λ 4 n in L 2 (0, 1), and using (4.18), we…”

Section: )mentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement

Guesmia

2017

Nonautonomous Dynamical Systems

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“…In [23] the authors considered the same problem as in [2] and they determined, when the condition (1.4) does not hold, the polynomial decay of solution with optimal decay rate to the system with mixed boundary condition. Other interesting results can be found in, for instance, [4,22,53,55,56,60].…”

Section: Introductionmentioning

confidence: 85%

Long-time dynamics of two classes of beam and plate equations

Monteiro¹

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Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions

Abdallah

Ghader

Wehbe

2018

Math Methods in App Sciences

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In this paper, we study the stability of a 1‐dimensional Bresse system with infinite memory‐type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. We consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly exponentially stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions.

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Asymptotic behavior to Bresse system with past history

Cited by 37 publications

References 17 publications

Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement

Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement

Long-time dynamics of two classes of beam and plate equations

Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions

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