On solvability of degenerate linear evolution equations with memory effects

By the methods of the operators semigroups theory a degenerate linear evolution equation with memory in a Banach space is reduced to a system of two equations. One of them is resolved with respect to the derivative, another has a nilpotent operator at the derivative. A problem with a given history for the first of the equations with memory is brought to the Cauchy problem for stationary equations system in a wider space. It allowed to obtain conditions of the unique solution existence for the problem, including solutions with a greater smoothness, by the methods of the classical operators semigroups theory. Thus unique solvability of the problem with a given history for a degenerate linear evolution equation with memory was researched with using some restrictions for the kernel of the memory integral operator. Besides, an analogous problem with generalized Showalter–Sidorov type condition on the history of the system was studied. General results were used for investigation of an initial boundary value problem for the linearized Oskolkov integro-differential system of equations, descibing the dynamics of the high order Kelvin–Voight fluid.