Asymptotic estimates related to an integro differential equation

The paper deals with an integrodi?erential operator which models numerous phenomena in superconductivity, in biology and in viscoelasticity. Initialboundary value problems with Neumann, Dirichlet and mixed boundary conditions are analyzed. An asymptotic analysis is achieved proving that for large t, the in uences of the initial data vanish, while the e?ects of boundary disturbances are everywhere bounded.