ABSTRACT
In this chapter, we consider some aspects of the qualitative theory of Sobolevtype equations. In Section 1, we introduce the Sobolev-Wiener spaces and establish embedding theorems. These theorems are used in Sections 2, 3 in the study of asymptotic properties of solutions to the mixed boundary-value problems in cylindrical domains. In Section 4, we study the asymptotic behavior of algebraic moments of the solution to the first boundary-value problem for the Sobolev equation as t→∞. We restrict ourselves to the first boundary-value problem for the Sobolev equation in the three-dimensional space. However, the results can be generalized to the n-dimensional case and the Sobolev system. Similar results can be obtained for the internal wave equation, the gravitygyroscopic wave equation, the Rossby wave equation, and so on. In Section 5, we study asymptotic properties of solutions the Cauchy problem for one equation appearing in the study of small-amplitude oscillations of a rotating compressible fluid. This equation is solved relative to the higher-order time-derivative. The asymptotic behavior as t→∞ of solutions to this equation essentially depends on the presence of the zero moments of the initial data. A similar situation holds for solutions to the Sobolev equation, the internal wave equations, and the Rossby wave equation (cf. S.V.Uspenskii, G.V.Demidenko, and V.G.Perepelkin [1] and S.V.Uspenskii and G.V.Demidenko [2]).
