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Investigation of Operator Models Arising in Viscoelasticity Theory

We study the correct solvability of initial-value problems for abstract integro-differential equations with unbounded operator coefficients in a Hilbert space. We undertake the spectral analysis of operator-functions that are symbols of such equations. The equations under consideration are an abstract form of linear integro-differential equations with partial derivatives, arising in viscoelasticity theory and various other important applications. We describe the localization and structure of the spectrum of operator-functions that are symbols of such equations.