A General Stability Result for Viscoelastic Equations with Singular Kernels

Messaoudi Sa Khenous Hb

doi:10.4172/2168-9679.1000221

Abstract

Viscoelasticity with regular relaxation functions has attracted the attention of many researchers over the last half century or so. Several results concerning existence and long-time behavior of solutions have been established. In particular the exponential, polynomial decay and recently what so called the general decay have been proved. For viscoelasticity, with singular kernels, less attention has been given and few results of existence and exponential decay have been established. In this paper we extend the general decay result, established for regularkernel viscoelasticity, to that with singular kernels. We also present some numerical test to illustrate our theoretical result.

Highlights

Since the pioneer work of Dafermos [1,2] in 1970, the viscoelastic equation t ∫ utt − ∆u +g(t − s)∆u(s)ds =0, (1.1)with smooth kernel, has attracted a great deal of researchers and several existence and stability results have been established
The author required the relaxation function to satisfy only restrictions deriving from Thermodynamics
He used the energy method to establish a stability theorem and obtained a regularity result for a class of singular kernels which ensures the asymptotic stability of the solution

Summary

Introduction

Since the pioneer work of Dafermos [1,2] in 1970, the viscoelastic equation t ∫ utt − ∆u +g(t − s)∆u(s)ds =0, (1.1)with smooth kernel, has attracted a great deal of researchers and several existence and stability results have been established. For instance, the works of [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] where the relaxation function was assumed to be either of polynomial or of exponential decay. He used the energy method to establish a stability theorem and obtained a regularity result for a class of singular kernels which ensures the asymptotic stability of the solution.

Results

Conclusion