Propagators of the Sobolev Equations

In this paper a initial-boundary value problem for the Sobolev equation is investigated. This problem is a part of more general mathematical model of wave propagation in uniform incompressible rotating with constant angular velocity $$\Omega $$ fluid. The studied problem may be obtained from it if we direct Oz axis collinear to $$\Omega $$. In addition an initial-boundary value problem for the Sobolev equation represents a model describing oscillations in stratified fluid. The solution of this problem is called an inertial (gyroscopic) wave because it arises by virtue of the Archimedes law and under influence of inertial forces. The paper shows that the relative spectrum of the pencil of operators entering the Sobolev equation is bounded. Then, based on the theory of relatively polynomially bounded pencils of operators and the theory of Sobolev type equations of higher order, the propagators of the Sobolev equation, given in a cylinder and in a parallelepiped are constructed.