Uniform decay properties of linear Volterra integro-differential equations

We establish some new results concerning the exponential decay and the polynomial decay of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation $$u_{tt}(t)- \alpha \Delta u(t) +\beta u_t(t) +\displaystyle\int_0^t\mu(s)\Delta u(t-s)ds=0$$ describing the motion of linearly viscoelastic solids. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel $\mu$.