Some unexplored questions arising in linear viscoelasticity

We consider the abstract integrodifferential equation modeling the dynamics of linearly viscoelastic solids. The equation is known to generate a semigroup S(t) on a certain phase space, whose asymptotic properties have been the object of extensive studies in the last decades. Nevertheless, some relevant questions still remain open, with particular reference to the decay rate of the semigroup compared to the decay of the memory kernel μ, and to the structure of the spectrum of the infinitesimal generator of S(t). This paper intends to provide some answers.