General decay properties of abstract linear viscoelasticity

Abstract

We consider a linear Volterra integro-differential equation of hyperbolic type, which can be viewed as an abstract version of the equation $$\begin{aligned} \partial _{tt} u(t)- \varDelta u(t) +\displaystyle \int \limits _0^t\mu (s)\varDelta u(t-s)\mathrm{d}s=0 \end{aligned}$$describing the motion of linearly viscoelastic solids. We establish some decay results for the associated energy, under assumptions that do not involve differential inequalities for the convolution kernel $$\mu $$.