Zeitschrift für angewandte Mathematik und Physik

, Volume 62, Issue 6, pp 1065–1082

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

Article

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 μ1, μ2 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.

Mathematics Subject Classification (2000)

35L05 35L15 35L70 93D15 

Keywords

Global existence General decay Relaxation function Delay feedbacks 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Mathématiques, Image et Applications Pole Sciences et TechnologieUniversité de la RochelleLa RochelleFrance
  2. 2.Department of Mathematics and StatisticsUniversity of KonstanzKonstanzGermany
  3. 3.Division of Mathematical and Computer Sciences and Engineering4700 King Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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