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Global Existence and Energy Decay of Solutions for a Nondissipative Wave Equation with a Time-Varying Delay Term

  • Abbes BenaissaEmail author
  • Salim A. Messaoudi
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)

Abstract

We consider the energy decay for a nondissipative wave equation in a bounded domain with a time-varying delay term in the internal feedback. We use an approach introduced by Guesmia which leads to decay estimates (known in the dissipative case) when the integral inequalities method due to Haraux-Komornik (Haraux in Nonlinear Partial Differential Equations and Their Applications. Collège de France seminar, Vol. VII (Paris, 1983–1984), pp. 161–179, 1985; Komornik in Exact Controllability and Stabilization: The Multiplier Method, 1994) cannot be applied due to the lack of dissipativity. First, we study the stability of a nonlinear wave equation of the form in a bounded domain. We consider the general case with a nonlinear function h satisfying a smallness condition and obtain the decay of solutions under a relation between the weight of the delay term in the feedback and the weight of the term without delay. We impose no control on the sign of the derivative of the energy related to the above equation. In the second case we take θconst and h(∇u)=−∇Φ⋅∇u. We prove an exponential decay result of the energy without any smallness condition on Φ.

Keywords

Nonlinear Wave Equation Decay Estimate Carleman Estimate Delay Term Observability Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We would like to thank very much the referees for their important remarks and comments which allow us to correct and improve this paper. This work has been partially funded by KFUPM under Project #FT111002.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Laboratory of Analysis and Control of Partial Differential EquationsDjillali Liabes UniversitySidi Bel AbbesAlgeria
  2. 2.Department of Mathematics and StatisticsKFUPMDhahranSaudi Arabia

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