Investigation of Integro-Differential Equations by Methods of Spectral Theory

This paper provides a survey of results devoted to the study of integro-differential equations with unbounded operator coefficients in a Hilbert space. These equations are operator models of integro-differential partial differential equations arising in numerous applications: in the theory of viscoelasticity, in the theory of heat propagation in media with memory (Gurtin–Pipkin equations), and in averaging theory. The most interesting and profound results of the survey are devoted to the spectral analysis of operator functions that are symbols of the integro-differential equations under study.