Global existence and decay estimates for the viscoelastic kirchhoff equation with a delay term

Abdelhamid Mohemmed Djaouti,Chafi Boudkhil,Noureddine Sebih

doi:10.22541/au.158231671.15211741

Abdelhamid Mohemmed Djaouti, Chafi Boudkhil + Show 1 more

Open Access

https://doi.org/10.22541/au.158231671.15211741

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Abstract

In this paper, we consider a viscoelastic kirchhoff equation with a delay term in the internal feedback. By using the Faedo-Galarkin approximation method we prove the well-posedness of the global solutions. Introducing suitable energy, we prove the general uniform decay results

Highlights

In this paper we investigate the global existence and uniform decay rate of the energy for solutions to the nonlinear viscoelastic kirchhof problem with delay term in the internal feedback
We introduce as in [19] a new variable z(x, ̺, t) = ut(x, t − τ), x ∈ Ω, ̺ ∈ (0, 1), t > 0
We will divide the proof into two steps: in the first step, we will use the FaedoGalerkin method to prove the existence of global solutions, where the second step is devoted to proving the uniform decay of the energy by the perturbed energy method

Summary

Introduction

In this paper we investigate the global existence and uniform decay rate of the energy for solutions to the nonlinear viscoelastic kirchhof problem with delay term in the internal feedback. Kirane and Said Houari in [12] investigated the following linear viscoelastic wave equation with a linear damping and delay term t utt − ∆u + g(t − s)∆u(s)ds + μ1ut + μ2ut(t − τ ) = 0 They showed that its energy was exponentially decaying when 0 < μ2 < μ1. [9] showed the energy decay of solutions for the following nonlinear viscoelastic equation with a time delay term in the internal feedback t utt + ∆2u − div (F (∇u)) − σ(t) g(t − s)∆2u(s)ds.

Assumptions and main result

Proof of the main result