A Study of Volterra Integro-Differential Equations Arising in Viscoelasticity Theory

Linear Volterra integro-differential equations with unbounded operator coefficients in a Hilbert space are studied. The integro-differential equations under consideration are operator models of problems in the theory of viscoelasticity, diffusion, and heat conduction in media with memory, and have a number of other important applications. A method is presented for reducing the initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to a Cauchy problem for a first-order differential equation in an extended function space. Using an approach based on operator semigroup theory, the correct solvability of the initial value problem for the Volterra integro-differential equation is established, along with corresponding estimates for the solution.