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∂_tt u(t)-α∆u(t)+β∂_t u(t)+∫_0^t μ(s)∆u(t-s)ds=0describing the motion of linearly viscoelastic solids. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel μ