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 Sobolev-type equations and obtaining conditions for their stability; the solvability of classes of problems for high-order Sobolev-type equations; the solvability and uniqueness of initial-finite 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 fundamental 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.