Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks

Ferhat Mohamed; Hakem Ali

doi:10.3934/dcdsb.2017024

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2017, Volume 22, Issue 2: 491-506. Doi: 10.3934/dcdsb.2017024

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Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks

Ferhat Mohamed and
Hakem Ali^,

Laboratory ACEDP, Djillali Liabes university, 22000 Sidi Bel Abbes, Algeria

* Corresponding author: hakemali@yahoo.com
* Corresponding author: hakemali@yahoo.com

Received: October 2015

Revised: October 2016

Published: December 2017

The authors are supported by CNEPRU-ALGERIA.

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Abstract

In this paper, we investigate the nonlinear wave equation in a bounded domain with a time-varying delay term in the weakly nonlinear internal feedback

$\left(|u_{t}|^{\gamma-2}u_{t}\right)_{t}-Lu-\int_{0}^{t}g(t-s)L u(s)ds+ \mu_{1} \psi(u_{t}(x, t))+ \mu_{2} \psi(u_{t}(x, t-\tau(t)))=0.$

The asymptotic behavior of solutions is studied by using an appropriate Lyapunov functional. Moreover, we extend and improve the previous results in the literature.

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Mathematics Subject Classification: Primary:35L05, 35L15;Secondary:93D15.

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