Decadimento uniforme per equazioni integro-differenziali lineari di Volterra

This talk is devoted to some recent results concerning the exponential and the polynomial decays 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 describing the motion of a linearly viscoelastic solid occupying a (bounded) volume at rest. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel. A similar analysis is carried on in the whole N-dimensional real space, although both the polynomial and the exponential decay of the memory kernel lead to a polynomial decay of the energy, with a rate influenced by the space dimension N. These results are contained in two joint papers with Monica Conti and Vittorino Pata (Politecnico di Milano).