Spectral analysis and solvability of Volterra integro-differential equations

Integro-differential equations with unbounded operator coefficients in a Hilbert space are studied. The equations under consideration are abstract hyperbolic equations perturbed by terms containing Volterra integral operators. These equations are operator models of integro-differential equations with partial derivatives arising in the theory of viscoelasticity, thermal physics, and homogenization problems in multiphase media. The correct solvability of these equations in weighted Sobolev spaces of vector functions is established, and a spectral analysis of the operator functions that are the symbols of these equations is carried out.