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Uniform Boundary Stabilization of the Wave Equation with a Nonlinear Delay Term in the Boundary Conditions

  • Wassila Ghecham
  • Salah-Eddine Rebiai
  • Fatima Zohra Sidiali
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

A wave equation in a bounded and smooth domain of \(\mathbb {R}^{n}\) with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established by adopting an approach due to Lasiecka and Tataru (Differ Integral Equ 6:507–533, 1993). The proof of existence of solutions relies on a construction of a suitable approximating problem for which the existence of solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behaviour of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Wassila Ghecham
    • 1
  • Salah-Eddine Rebiai
    • 1
  • Fatima Zohra Sidiali
    • 1
  1. 1.LTMUniversity of Batna 2BatnaAlgeria

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