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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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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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Mathematics Subject Classification (2000)

  • 35L05
  • 35L15
  • 35L70
  • 93D15

Keywords

  • Global existence
  • General decay
  • Relaxation function
  • Delay feedbacks