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Abstract
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 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.
Highlights
The Bresse system [4], known as the circular arch problem, is the following coupled three hyperbolic equations: ⎧ ⎪ρ1φtt − kx − lk0 = F1 in (0, L) × (0, ∞), ⎪ ⎨ρ2ψtt − bψxx + k = F2 in (0, L) × (0, ∞), (1.1)⎪ ⎪ ⎩ ρ1wtt − k0x + lk = F3 in (0, L) × (0, ∞), where ρ1, ρ2, b, k, k0, l and L are positive constants, Fj : (0, L) × (0, ∞) → R, j = 1, 2, 3, are given external forces, which play the role of controls, and φ, ψ and w represent, respectively, the vertical, shear angle and longitudinal displacements
When only one frictional damping is considered on the longitudinal or shear angle displacement, it was proved that the exponential stability
Similar stability results to the ones of [11] were proved in [9] under one infinite memory acting on the longitudinal displacement with kernels having a general decay at infinity, and in [6] 20 under one infinite memory acting on the shear angle displacement with kernels having an esponential decay at infinity
Summary
The Bresse system [4], known as the circular arch problem, is the following coupled three hyperbolic equations:. When only one frictional damping is considered on the longitudinal or shear angle displacement (that is γ1 = γ2 = 0 or γ1 = γ3 = 0), it was proved that the exponential stability. Similar stability results were proved in [1], [8], [13], [17] and [18] in case where the Bresse system is coupled with one or two heat equations in a certain manner so that at least the longitudinal or shear angle displacement is indirectely damped via the heat equations. When only two memories are considered, the stability of Bresse systems was proved in [11], where the decay rate depends on si and on the smoothness of initial data.
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