Operator Approach to the Ilyushin Model for a Viscoelastic Body of Parabolic Type

Abstract

The problem of small movements of a viscoelastic body of parabolic type is studied in the paper. The unique strong solvability of the corresponding initial-boundary value problem is proved. The spectrum and the properties of root elements of the emerging operator block are studied. More precisely, the theorem about both the essential and the discrete spectrum of the main operator block is proved. The asymptotic formula for the series of eigenvalues condensing at infinity is found. Completeness and the basis property of the system of root elements of the main operator are established. Presentations for a solution of the original second-order integrodifferential equation are found both in the form of contour integrals and expansions in the system of eigenvectors of some operator pencil. A certain statement concerning stabilization of solutions to the evolution problem is proved. In the last section, the case of a synchronously isotropic medium of parabolic type is studied as a particular case of the model considered.