General decay for a system of viscoelastic wave equation with past history, distributed delay and Balakrishnan-Taylor damping terms

Abdelbaki Choucha; Salah Boulaaras; Djamel Ouchenane; Salem Alkhalaf; Rashid Jan; Abdelbaki Choucha; Salah Boulaaras; Djamel Ouchenane; Salem Alkhalaf; Rashid Jan

doi:10.3934/era.2022199

Electronic Research Archive

2022, Volume 30, Issue 10: 3902-3929. doi: 10.3934/era.2022199

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General decay for a system of viscoelastic wave equation with past history, distributed delay and Balakrishnan-Taylor damping terms

1.
Department of Mathematics, Faculty of Exact Sciences, University of El Oued, Algeria
2.
Department of Material Sciences, Faculty of Sciences, Amar Teleji Laghouat University, Algeria
3.
Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia
4.
Department of Mathematics, Faculty of Sciences, Amar Teleji Laghouat University, Algeria
5.
Department of Computer Sciences, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia
6.
Department of Mathematics, University of Swabi, Swabi 23561, Pakistan

Received: 25 June 2022 Revised: 12 August 2022 Accepted: 22 August 2022 Published: 29 August 2022

The subject of this research is a coupled system of nonlinear viscoelastic wave equations with distributed delay components, infinite memory and Balakrishnan-Taylor damping. Assume the kernels $ g_{i} :{\bf R}_{+}\rightarrow {\bf R}_{+} $ holds true the below

$ g_{i}'(t)\leq-\zeta_{i}(t)G_{i}(g_{i}(t)), \quad \forall t\in {\bf R}_{+}, \quad {\rm{for}} \quad i = 1, 2, $

in which $ \zeta_{i} $ and $ G_{i} $ are functions. We demonstrate the stability of the system under this highly generic assumptions on the behaviour of $ g_i $ at infinity and by dropping the boundedness assumptions in the historical data.
- wave equation,
- infinite memory,
- distributed delay term,
- viscoelastic term,
- relaxation function
Citation: Abdelbaki Choucha, Salah Boulaaras, Djamel Ouchenane, Salem Alkhalaf, Rashid Jan. General decay for a system of viscoelastic wave equation with past history, distributed delay and Balakrishnan-Taylor damping terms[J]. Electronic Research Archive, 2022, 30(10): 3902-3929. doi: 10.3934/era.2022199

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Abstract

The subject of this research is a coupled system of nonlinear viscoelastic wave equations with distributed delay components, infinite memory and Balakrishnan-Taylor damping. Assume the kernels $ g_{i} :{\bf R}_{+}\rightarrow {\bf R}_{+} $ holds true the below

$ g_{i}'(t)\leq-\zeta_{i}(t)G_{i}(g_{i}(t)), \quad \forall t\in {\bf R}_{+}, \quad {\rm{for}} \quad i = 1, 2, $

in which $ \zeta_{i} $ and $ G_{i} $ are functions. We demonstrate the stability of the system under this highly generic assumptions on the behaviour of $ g_i $ at infinity and by dropping the boundedness assumptions in the historical data.

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