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

100%
Guesmia A.
Nonautonomous Dynamical Systems
|
2017
|
tom 4
|
nr 1
78-97
EN
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 ofwave propagations are the same, and to the polynomial one with decay rate [...] 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 [...] . 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.
2
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On the nonlinear stabilization of the wave equation

100%
Guesmia A.
Annales Polonici Mathematici
|
1998
|
tom 68
|
nr 2
191-198
EN
We obtain a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with nonlinear internal and boundary feedbacks. We show that a judicious choice of the feedbacks leads to fast energy decay.
3
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Decay estimates of solutions of a nonlinearly damped semilinear wave equation

63%
Guesmia A., Messaoudi S.
Annales Polonici Mathematici
|
2005
|
tom 85
|
nr 1
25-36
EN
We consider an initial boundary value problem for the equation $u_{tt} - Δu - ∇ϕ·∇u + f(u) + g(u_{t}) = 0$. We first prove local and global existence results under suitable conditions on f and g. Then we show that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. This result improves and includes earlier decay results established by the authors.
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