# Sobolev-Type Systems and Applied Problems

A.V. KellerThis 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 six 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.Full text

- Keywords
- Sobolev-type equations; G.A. Sviridyuk's phase space method; degenerate resolving (semi)groups; Showalter-Sidorov condition; initial-final conditions; optimal control.
- References
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91. Zagrebina S.A., Soldatova E.A., Sviridyuk G.A. The Stochastic Linear Oskolkov Model of the Oil Transportation by the Pipeline. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 317-325. DOI: 10.1007/978-3-319-12145-1_20

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94. Zamyshlyaeva A.A., Al-Isawi J.K.T. On Some Properties of Solutions to One Class of Evolution Sobolev Type Mathematical Models in Quasi-Sobolev Spaces. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 4, pp. 113-119. DOI: 10.14529/mmp150410

95. Zamyshlyaeva A.A., Bychkov E.V. The Cauchy Problem for the Sobolev Type Equation of Higher Order. Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 1, pp. 5-14. DOI: 10.14529/mmp180101

96. Zamyshlyaeva A., Lut A. Inverse Problem for the Sobolev Type Equation of Higher Order. Mathematics, 2021, vol. 9, no. 14, p. 1647. DOI: 10.3390/math9141647

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98. Zamyshlyaeva A.A., Sviridyuk G.A. Nonclassical Equations of Mathematical Physics. Linear Sobolev Type Equations of Higher Order. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 4, pp. 5-16. DOI: 10.14529/mmph160401

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