On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions

Tae Gab Ha

doi:10.3934/eect.2018014

Abstract

In this paper, we consider the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions. This work is devoted to prove, under suitable conditions on the initial data, the global existence and uniform decay rate of the solutions when the relaxation function is not necessarily of exponential or polynomial type.

Highlights

In this paper, we are concerned with the asymptotic stability of the viscoelastic equation: u − M (t)∆u + t 0 h(t − s)∆u(s)ds = |u|ρu u = 0 M (t) ∂u ∂ν s)(s)ds k(x)z u + f (x)z + q(x)z = 0u(x, 0) = u0(x), u (x, 0) = u1(x) z(x, 0)
The goal of this paper is to study the asymptotic stability of the nonlinear viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions by adopting and modifying the perturbed energy technique based on the work in [26]
T 0 h(t s)(u(t) u(s))dsdΓ; By using Young’s inequality and the trace imbedding theorem, we deduce that k1 ||k1/2z 4δ (t)||22,Γ1

Summary

Introduction

Viscoelastic equation, acoustic boundary conditions, BalakrishnanTaylor damping, global existence, general decay rate. Frota and Larkin [11] considered global solvability and the exponential decay of the energy for the wave equation with acoustic boundary conditions, which was eliminated the second derivative term for 1 ≤ n ≤ 3. On viscoelastic type problem with acoustic boundary conditions, there were very few results.

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