Asymptotic behavior for a viscoelastic Kirchhoff equation with distributed delay and Balakrishnan\u2013Taylor damping

Abdelbaki Choucha,Salah Boulaaras

doi:10.1186/s13661-021-01555-0

Abstract

A nonlinear viscoelastic Kirchhoff-type equation with Balakrishnan–Taylor damping and distributed delay is studied. By the energy method we establish the general decay rate under suitable hypothesis.

Highlights

Let H = × (τ1, τ2) × (0, ∞), in the present work, we consider the following Kirchhoff equation: ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨|ut |p + +utt – (ζ0 + ζ1 α(t) t 0 h(t τ2 τ1 |β2(s)||ut ∇u + σ
See [4–6, 9–11, 13–18, 22, 29, 31, 32, 34, 35]. It has been studied by many authors, especially in Kirchhoff ’s equations
Our paper is divided into several sections: we lay down the hypotheses, concepts, and lemmas we need, and in the last section we prove our main result

Summary

Introduction

Represents distributive time delay, h, α are positive functions. It has been studied by many authors, especially in Kirchhoff ’s equations (see [8, 10, 19–21, 23–26, 30, 33]). The stability and the asymptotic behavior of evolution systems with time delay, especially the distributed delay effect, have been studied by many authors. 2 Preliminaries For studying our problem, we will need some materials.

Results

Conclusion