Well-posedness of Volterra integro-differential equations with singular kernels

This paper is devoted to the study of integro-differential equations with unbounded operator coefficients in a Hilbert space. The principal part of the equations considered is an abstract hyperbolic equation perturbed by terms containing Volterra integral operators with singular kernels. Such integro-differential equations describe, for example, linear viscoelastic phenomena, heat propagation processes in media with memory (the Gurtin–Pipkin equation), averaging problems in perforated media (Darcy's law), etc. We establish conditions for the existence and uniqueness of strong and generalized solutions to initial-boundary-value problems for these equations in weighted Sobolev spaces on the positive semiaxis.