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Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weight

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Abstract

In this research work, we consider the second-order viscoelastic equation with a weak internal damping, a time-varying delay term and nonlinear weights

$$\begin{aligned} u_{tt}(t) + {\mathcal {A}} u(t) - \int _0^t g (t-s) {\mathcal {A}} u(s) ds + \mu _1 (t) u_t (t)+ \mu _2(t) u_t(t-\tau (t)) =0 \forall t>0 , \end{aligned}$$

together with suitable initial conditions. We first prove the existence of a unique global weak solution by means of the classical Faedo–Galerkin method. Then, by assuming the general condition:

$$\begin{aligned} g'(t) \le - \xi (t) H(g(t)), \forall t\ge 0, \end{aligned}$$

where H is a positive increasing and convex function and \(\xi \) is a positive function which is not necessarily monotone, we establish optimal explicit and general stability estimates which rely on the well-known multipliers method and some properties of convex functions. This study generalizes and improves many earlier ones in the existing literature.

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Acknowledgements

The authors thank the referees for their useful remarks and suggestions which helped them to improve the presentation of the paper. This work is supported by D.G.R.S.D.T.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Exact Sciences, Mustapha Stambouli University, P. O. Box 305, 29000, Mascara, Algeria

    Hocine Makheloufi & Mounir Bahlil

  2. Laboratory of Analysis and Control of Partial Differential Equations, Djillali Liabes University, P. O. Box 89, 22000, Sidi Bel Abbes, Algeria

    Hocine Makheloufi & Mounir Bahlil

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  1. Hocine Makheloufi
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Correspondence to Hocine Makheloufi.

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Makheloufi, H., Bahlil, M. Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weight. Ricerche mat 73, 433–469 (2024). https://doi.org/10.1007/s11587-021-00617-w

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