LAUSR

We study abstract integro-differential equations that are operator models of problems in viscoelasticity. We present results based on an approach related to the study of one-parameter semigroups for linear evolution equations. The presented approach can also be used to study other integro-differential equations containing integral terms of the Volterra convolution form. A method is given for reducing the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation. The existence of a contraction $$C_0 $$ -semigroup is proved under certain assumptions about the kernels of integral operators. Examples of exponential and fractional-exponential (Rabotnov functions) kernels of integral operators satisfying the above assumptions are given.