The asymptotic behavior of the Bresse–Cattaneo system

Said‐Houari, Belkacem; Hamadouche, Taklit

doi:10.1142/s0219199715500455

Cited by 19 publications

(10 citation statements)

References 19 publications

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“…where

U = {false(ϕ_{x} - ψ - l w, ϕ_{t}, a ψ_{x}, ψ_{t}, k_{0} ((), w_{x} - l ϕ), w_{t}, θ, q false)}^{T}

and α is the parameter

α (τ_{q}) = α = (τ_{q} - 1) (1 - a^{2}) - τ_{q} m^{2} .

No optimality of the decay rates has been shown in Said‐Houari and Hamadouche . Similar decay rates have been also proved for the Bresse‐Fourier system (the system associated to τ q =0) in Said‐Houari and Soufyane under the same assumption on α (0).…”

Section: Introductionmentioning

confidence: 57%

“…Recently, Said‐Houari and Hamadouche investigated the decay rate of system and showed the following: •For α =0,

false‖ \partial_{x}^{k} U false(t false) ‖_{L^{2}} \leq C {false(1 + t false)}^{- 1 false/ 12 - k false/ 6} false‖ U_{0} ‖_{L^{1}} + C {false(1 + t false)}^{- ℓ false/ 2} false‖ \partial_{x}^{k + ℓ} U_{0} ‖_{L^{2}} .

•For α ≠0,

false‖ \partial_{x}^{k} U false(t false) ‖_{L^{2}} \leq C {false(1 + t false)}^{- 1 false/ 12 - k false/ 6} false‖ U_{0} ‖_{L^{1}} + C {false(1 + t false)}^{- ℓ false/ 10} false‖ \partial_{x}^{k + ℓ} U_{0} ‖_{L^{2}},

…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new stability number of the Bresse‐Cattaneo system

Djouamai

Said‐Houari

2018

Math Methods in App Sciences

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In this paper, we consider the Bresse‐Cattaneo system with a frictional damping term and prove some optimal decay results for the L2‐norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number δ that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results

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“…where

U = {false(ϕ_{x} - ψ - l w, ϕ_{t}, a ψ_{x}, ψ_{t}, k_{0} ((), w_{x} - l ϕ), w_{t}, θ, q false)}^{T}

and α is the parameter

α (τ_{q}) = α = (τ_{q} - 1) (1 - a^{2}) - τ_{q} m^{2} .

Section: Introductionmentioning

confidence: 57%

“…Recently, Said‐Houari and Hamadouche investigated the decay rate of system and showed the following: •For α =0,

false‖ \partial_{x}^{k} U false(t false) ‖_{L^{2}} \leq C {false(1 + t false)}^{- 1 false/ 12 - k false/ 6} false‖ U_{0} ‖_{L^{1}} + C {false(1 + t false)}^{- ℓ false/ 2} false‖ \partial_{x}^{k + ℓ} U_{0} ‖_{L^{2}} .

•For α ≠0,

false‖ \partial_{x}^{k} U false(t false) ‖_{L^{2}} \leq C {false(1 + t false)}^{- 1 false/ 12 - k false/ 6} false‖ U_{0} ‖_{L^{1}} + C {false(1 + t false)}^{- ℓ false/ 10} false‖ \partial_{x}^{k + ℓ} U_{0} ‖_{L^{2}},

…”

Section: Introductionmentioning

confidence: 99%

A new stability number of the Bresse‐Cattaneo system

Djouamai

Said‐Houari

2018

Math Methods in App Sciences

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“…Under suitable relations between the constants, the authors of [16] showed exponential and optimal polynomial decay rates. The same system was treated by Said-Houari and Hamadouche [25] in the whole space R, where they showed that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and they proved that the decay rate of the solution in the L 2 -norm is of the rate t −1/12 . For more problems of thermoelastic Bresse systems, we refer the reader to [24], where a global existence was proved using two heat equations, and to [26,27], where Cauchy thermoelastic Bresse problems were treated.…”

Section: Introductionmentioning

confidence: 83%

Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements

2021

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“…e coupling between the Fourier law of heat condition and different systems has been considered by many authors and there are many results. For example, Bresse system (Bresse-Fourier) has been studied in [11], the Bresse system coupled with the Cattaneo law of heat condition (Bresse-Cattaneo) has been studied in [12], and Timoshenko system with past history has been considered in [13]. For further details, we refer the reader to the following papers [14,15].…”

Section: Introductionmentioning

confidence: 99%