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Existence and asymptotic stability of a viscoelastic wave equation with a delay

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Abstract

In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem

$$u_{tt}(x,t)-\Delta u(x,t)+\int\limits_{0}^{t}g(t-s){\Delta}u(x,s){d}s+\mu_{1}u_{t}(x,t)+\mu_{2} u_{t}(x,t-\tau)=0$$

together with initial conditions and boundary conditions of Dirichlet type. Here \({(x,t)\in\Omega\times (0,\infty), g}\) is a positive real valued decreasing function and μ 1, μ 2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo–Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals.

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Author notes

  1. Belkacem Said-Houari

    Present address: Division of Mathematical and Computer Sciences and Engineering, 4700 King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

Authors and Affiliations

  1. Mathématiques, Image et Applications Pole Sciences et Technologie, Université de la Rochelle, La Rochelle, France

    Mokhtar Kirane

  2. Department of Mathematics and Statistics, University of Konstanz, 78457, Konstanz, Germany

    Belkacem Said-Houari

Authors
  1. Mokhtar Kirane
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  2. Belkacem Said-Houari
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Correspondence to Belkacem Said-Houari.

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Kirane, M., Said-Houari, B. Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 62, 1065–1082 (2011). https://doi.org/10.1007/s00033-011-0145-0

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  • DOI: https://doi.org/10.1007/s00033-011-0145-0

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