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