General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term

Wenjun Liu; Biqing Zhu; Gang Li; Danhua Wang

doi:10.3934/eect.2017013

Article Contents

2017, Volume 6, Issue 2: 239-260. Doi: 10.3934/eect.2017013

This issue Previous Article Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions Next Article Viscoelastic plate equation with boundary feedback

General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author:Wenjun Liu
* Corresponding author:Wenjun Liu

Received: February 16, 2016

Revised: January 28, 2017

Published: May 2017

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In this paper, we consider a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term acting on the boundary. By using the Faedo-Galerkin approximation method, we first prove the well-posedness of the solutions. By introducing suitable energy and perturbed Lyapunov functionals, we then prove the general decay results, from which the usual exponential and polynomial decay rates are only special cases. To achieve these results, we consider the following two cases according to the coefficient α of the strong damping term: for the presence of the strong damping term (α>0), we use the strong damping term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the strong damping term; for the absence of the strong damping term (α=0), we use the viscoelasticity term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the kernel function.

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Mathematics Subject Classification: Primary: 35L05; Secondary: 35L20, 35L70, 93D15.

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