Arbitrary decays for a viscoelastic equation

Shun-Tang Wu

doi:10.1186/1687-2770-2011-28

Abstract

In this paper, we consider the nonlinear viscoelastic equation ∣ u t ∣ ρ u t t -Δu-Δ u t t + ∫ 0 t g ( t - s ) Δu ( s ) d s+∣u ∣ p u=0, in a bounded domain with initial conditions and Dirichlet boundary conditions. We prove an arbitrary decay result for a class of kernel function g without setting the function g itself to be of exponential (polynomial) type, which is a necessary condition for the exponential (polynomial) decay of the solution energy for the viscoelastic problem. The key ingredient in the proof is based on the idea of Pata (Q Appl Math 64:499-513, 2006) and the work of Tatar (J Math Phys 52:013502, 2010), with necessary modification imposed by our problem.Mathematical Subject Classification (2010): 35B35, 35B40, 35B60

Highlights

1 Introduction It is well known that viscoelastic materials have memory effects
From the mathematical point of view, their memory effects are modeled by an integro-differential equations
This type of equations usually arise in the theory of viscoelasticity when the material density varies according to the velocity. They proved a global existence result of weak solutions for g ≥ 0 and a uniform decay result for g > 0

Summary

Introduction

It is well known that viscoelastic materials have memory effects. These properties are due to the mechanical response influenced by the history of the materials themselves. This type of equations usually arise in the theory of viscoelasticity when the material density varies according to the velocity In that paper, they proved a global existence result of weak solutions for g ≥ 0 and a uniform decay result for g > 0. Messaoudi and Tatar [14] studied problem (1.1) for the case of g = 0, they improved the result in [3] by showing that the solution goes to zero with an exponential or polynomial rate, depending on the decay rate of the relaxation function g. More recently Tatar [25] investigated the asymptotic behavior to problem (1.1) with r = g = 0 when h(t)g(t) Î L1(0, ∞) for some nonnegative function h(t) He generalized earlier works to an arbitrary decay not necessary of exponential or polynomial rate.

Preliminary results

Methods