SOBOLEV-TYPE SYSTEMS AND APPLIED PROBLEMS

This article provides a brief overview of analytical studies of Sobolev-type equations obtained by the research team at the South Ural State University. The review includes results in the areas: the solvability of initial problems for linear and semi-linear Sobolevtype equations and obtaining conditions for their stability; the solvability of classes of problems for high-order Sobolev-type equations; the solvability and uniqueness of initialfinite problems and optimal control problems for Sobolev-type equations; the theory of stochastic Sobolev-type equations; the solvability of problems for Sobolev-type equations in the space of K-forms. The results are based on the use of the phase-space method and the theory of degenerate resolving (semi)groups developed by Sviridyuk and his students. Sobolev-type equations are the basis of various physical, biological, and economic models, a summary of the results of this area of research gives a systematic up-to-date understanding of it. The article contains five sections, the bibliography of the review includes fundamen-tal works that have become the basis for many subsequent results, primarily numerical studies, and recent works expanding the methods and theory of Sobolev-type equations.