To the Problem of Small Oscillations of a System of Two Viscoelastic Fluids Filling Immovable Vessel: Model Problem

In this paper, we study the scalar conjugation problem, which models problem of small oscillations two viscoelastic fluids filling a fixed vessel. An initial-boundary value is investigated and theorem on its unique solvability positive semiaxis proven with semigroup theory methods. The spectral that arises in case for normal system studied by methods operator functions (operator pencils). resulting pencil generalizes both well-known Kreyn’s (oscillations viscous fluid an open vessel) arising motions partially filled example two-dimensional allowing separation variables considered, all points essential spectrum branches eigenvalues are found. Based hypothesis structure formulated multiple basis property root elements main proved.