Sobolev-type systems and applied problems

A. V. Keller

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 16 (2023) no. 4, pp. 5-32

Résumé

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.

Détail

Mots-clés : Sobolev-type equations, initial-final conditions
Keywords: G.A. Sviridyuk's phase space method, degenerate resolving (semi)groups, Showalter–Sidorov condition, optimal control.

Bibliographie

[1] Al'shin A.B., Korpusov M.O., Sveshnikov A.G., Blow-Up in Nonlinear Sobolev Type Equations, Walter de Gruyter, Berlin, 2011 | DOI | MR

[2] Banasiak J., Manakova N.A., Sviridyuk G.A., “Positive Solutions to Sobolev Type Equations with Relatively $p$-sectorial Operators”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 13:2 (2020), 17–32 | DOI

[3] Bokareva T.A., Sviridyuk G.A., “Whitney Folds in Phase Spaces of Some Semilinear Sobolev-Type Equations”, Mathematical Notes, 55:3 (1994), 237–242 | DOI | MR | Zbl

[4] Boyarintsev Y.E., Chistyakov V.F., Algebraic Differential Systems: Methods of Soluthin and Research, Nauka, Novosibirsk, 1998 (in Russian) | MR

[5] Bychkov E.V., “Analytical Study of the Mathematical Model of Wave Propagation in Shallow Water by the Galerkin Method”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 14:1 (2021), 26–38 | DOI | Zbl

[6] Buevich A.V., Sagadeeva M.A., Zagrebina S.A., “Stabilitty of a Stationary Solution to One Class of Non–Autonomous Sobolev Type Equations”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 16:3 (2023), 65–73 | DOI

[7] Chistyakov V.F., Shcheglova A.A., Selected Chapters of Theory of Algebro-Differential Systems, Siberian Publishing House Nauka, Novosibirsk, 2003 (in Russian) | MR

[8] Demidenko G.V., Uspenskii S.V., Partial Differential Equations and Systems not Solvable with Respect to the Highest Order Derivative, Marcel Dekker, N.Y.–Basel–Hong Kong, 2003 | MR | Zbl

[9] Favini A., Sviridyuk G.A., Manakova N.A., “Linear Sobolev Type Equations with Relatively $p$-Sectorial Operators in Space of “Noises””, Abstract and Applied Analysis, 2015, 697410 | DOI | MR | Zbl

[10] Favini A., Sviridyuk G., Sagadeeva M., “Linear Sobolev Type Equations with Relatively $p$-Radial Operators in Space of “Noises””, Mediterranean Journal of Mathematics, 13:6 (2016), 4607–4621 | DOI | MR | Zbl

[11] Favini A., Zagrebina S.A., Sviridyuk G.A., “Multipoint Initial-Final Value Problems for Dynamical Sobolev-Type Equations in the Space of Noises”, Electronic Journal of Differential Equations, 2018 (2018), 128, 10 pp. | MR | Zbl

[12] Favini A., Sviridyuk G.A., Zamyshlyaeva A.A., “One Class of Sobolev Type Equations of Higher Order with Additive “White Noise””, Communications on Pure and Applied Analysis, 15:1 (2016), 185–196 | DOI | MR | Zbl

[13] Gavrilova O.V., “Numerical Study on the Non-Uniqueness of Solutions to the Showalter–Sidorov Problem for One Degenerate Mathematical Model of an Autocatalytic Reaction with Diffusion”, Journal of Computational and Engineering Mathematics, 6:4 (2019), 3–17 | DOI | MR | Zbl

[14] Gil'mutdinova A.F., “On the Non-Uniqueness of Solutions of Showalter–Sidorov Problem for One Plotnikov Model”, Vestnik of Samara State University, 2007, no. 9/1, 85–90 (in Russian) | MR | Zbl

[15] Gliklikh Yu. E., Mashkov E. Yu., “Stochastic Leontieff Type Equations in Terms of Current Velocities of the Solution II”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 9:3 (2016), 31–40 | DOI | Zbl

[16] Goncharov N.S., Zagrebina S.A., Sviridyuk G.A., “The Non-Uniqueness of the Showalter–Sidorov Problem for the Barenblatt–Zheltov–Kochina Equation with Wentzell Boundary Conditions in a Bounded Domain”, Book of Abstracts of O.A. Ladyzhenskaya Centennial Conference on PDE's (St. Petersburg, 2022), 56

[17] Goncharov N.S., Sviridyuk G.A., “Analysis of the Stochastic Wentzell System of Fluid Filtration Equations in a Circle and on its Boundary”, Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 15:3 (2023), 15–22 | DOI

[18] Keller A.V., “On the Computational Efficiency of the Algorithm of the Numerical Solution of Optimal Control Problems for Models of Leontieff Type”, Journal of Computational and Engineering Mathematics, 2:2 (2015), 39–59 | DOI | Zbl

[19] Keller A.V., “Leontief-Type Systems and Applied Problems”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 15:1 (2022), 23–42 | DOI | Zbl

[20] Keller A.V., Al-Delfi J.K., “Holomorphic Degenerate Groups of Operators in Quasi–Banach Spaces”, Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 7:1 (2015), 20–27 | Zbl

[21] Kitaeva O.G., “Invariant Manifolds OF Semilinear Sobolev Type Equations”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 15:1 (2022), 101–111 | DOI | Zbl

[22] Kitaeva O.G., Shafranov D.E., Sviridyuk G.A., “Degenerate Holomorphic Semigroups of Operators in Spaces of K-“Noise” on Riemannian Manifolds”, Semigroups of Operators – Theory and Applications, 325 (2020), 279–292 | DOI | MR | Zbl

[23] Kitaeva O.G., Shafranov D.E., Sviridyuk G.A., “Exponential Dichotomies in Barenblatt–Zheltov–Kochina Model in Spaces of Differential Forms with “Noise””, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 12:2 (2019), 47–57 | DOI | MR | Zbl

[24] Korpusov M.O., Panin A.A., Shishkov A.E., “On the Critical Exponent “Instantaneous Blow-up” Versus “Local Solubility” in the Cauchy Problem for a Model Equation of Sobolev Type”, Izvestiya RAN: Mathematics, 85:1 (2021), 111–144 | DOI | MR | Zbl

[25] Konkina A.S., “Multipoint Initial-Final Value Problem for the Model of Davis with Additive White Noise”, Bulletin of the South Ural State University. Mathematical Modelling, Programming and Computer Software, 10:2 (2017), 144–149 | DOI

[26] Kozhanov A.I., “Boundary Value Problems for Fourth-Order Sobolev Type Equations”, Journal of Siberian Federal University. Mathematics and Physics, 14:4 (2021), 425–432 | DOI | MR

[27] Kozhanov A.I., “On a Nonlocal Boundary Value Problem with Variable Coefficients for the Heat Equation and the Aller Equation”, Differential Equations, 40:6 (2004), 815–826 | DOI | MR | Zbl

[28] Manakova N.A., Gavrilova O.V., Perevozchikova K.V., “Semilinear Models of Sobolev Type. Non-Uniqueness of Solution to the Showalter–Sidorov Problem”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 15:1 (2022), 84–100 | DOI | Zbl

[29] Manakova N.A., Sviridyuk G.A., “An Optimal Control of the Solutions of the Initial-Final Problem for Linear Sobolev Type Equations with Strongly Relatively $p$-Radial Operator”, Springer Proceedings in Mathematics and Statistics, 113, 2015, 213–224 | DOI | MR | Zbl

[30] Manakova N.A., “Mathematical Models and Optimal Control of the Filtration and Deformation Processes”, Bulletin of the South Ural State University. Mathematical Modelling, Programming and Computer Software, 8:3 (2015), 5–24 | DOI | Zbl

[31] Manakova N.A., “An Optimal Control to Solutions of the Showalter–Sidorov Problem for the Hoff Model on the Geometrical Graph”, Journal of Computational and Engineering Mathematics, 1:1 (2014), 26–33 | MR | Zbl

[32] Manakova N.A., Sviridyuk G.A., “Non-Classical Equations of Mathematical Physics. Phase Spaces of Semilinear Sobolev Equations”, Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 8:3 (2016), 31–51 | DOI | Zbl

[33] Manakova N.A., Gavrilova O.V., “About Nonuniqueness of Solutions of the Showalter–Sidorov Problem for One Mathematical Model of Nerve Impulse Spread in Membrane”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 11:4 (2018), 161–168 | DOI | MR | Zbl

[34] Manakova N.A., Perevozhikova K.V., “Numerical Simulation Of Start Control and Final Observation in Fluid Filtration Model”, Journal of Computational and Engineering Mathematics, 8:1 (2021), 29–45 | DOI | MR

[35] Melnikova I.V., Filinkov A.I., The Cauchy Problem: Three Approaches, Chapman, London; Hall/CRC, N.Y., 2001 | MR

[36] Pyatkov S.G., “Boundary Value and Inverse Problems for Some Classes of Nonclassical Operator-Differential Equations”, Siberian Mathematical Journal, 62:3 (2021), 489–502 | DOI | MR | Zbl

[37] Pyatkov S.G., Operator Theory. Nonclassical Problems, VSP, Utrecht–Boston–Köln–Tokyo, 2002 | DOI | MR | Zbl

[38] Sagadeeva M.A., Rashid A.S., “Existence of Solutions in Quasi–Banach Spaces for Evolutionary Sobolev Type Equations in Relatively Radial Case”, Journal of Computational and Engineering Mathematics, 2:2 (2015), 71–81 | DOI | MR | Zbl

[39] Sagadeeva M.A., “Degenerate Flows of Solving Operators for Nonstationary Sobolev Type Equations”, Bulletin of the South Ural State University. Mathematics. Mechanics. Physics, 9:1 (2017), 22–30 | DOI | Zbl

[40] Sagadeeva M.A., Zagrebina S.A., Manakova N.A., “Optimal Control of Solutions of a Multipoint Initial-Final Problem for Non-Autonomous Evolutionary Sobolev Type Equation”, Evolution Equations and Control Theory, 8:3 (2019), 473–488 | DOI | MR | Zbl

[41] Shafranov D.E., “Numerical Solution of the Hoff Equation with Additive “White Noise” in Spaces of Differential Forms on a Torus”, Journal of Computational and Engineering Mathematics, 8:2 (2021), 46–55 | DOI | MR | Zbl

[42] Shafranov D.E., Kitaeva O.G., “The Barenblatt–Zheltov–Kochina Model with the Showalter–Sidorov Condition and Additive “White Noise” in Spaces of Differential Forms on Riemannian Manifolds without Boundary”, Global and Stochastic Analysis, 5:2 (2018), 145–159

[43] Shafranov D.E., Kitaeva O.G., Sviridyuk G.A., “Stochastic Equations of Sobolev Type with Relatively $p$-Radial Operators in Spaces of Differential Forms”, Differential Equations, 57:4 (2021), 507–516 | DOI | MR | Zbl

[44] Shestakov A.L., Sviridyuk G.A., “Optimal Measurement of Dynamically Distorted Signals”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 17(234) (2011), 70–75 | Zbl

[45] Shestakov A. L., Sviridyuk G.A., Zamyshlyaeva A.A., Keller A.V., Khudyakov Y.V., “Numerical Investigation of Optimal Dynamic Measurements”, Acta IMEKO, 7:2 (2018), 65–72 | DOI

[46] Shestakov A. L., Keller A.V., “Optimal Dynamic Measurement Method Using Digital Moving Average Filter”, Journal of Physics: Conference Seriesthis, 1864:1 (2021), 012073 | DOI | MR

[47] Showalter R.E., “The Sobolev type Equations”, Applicable Analysis, 5:1 (1975), 15–22 | DOI | MR | Zbl

[48] Showalter R.E., “The Sobolev type Equations. II”, Applicable Analysis, 5:2 (1975), 81–99 | DOI | MR | Zbl

[49] Sidorov N., Loginov B., Sinithyn A., Falaleev M., Lyapunov–Shmidt Methods in Nonlinear Analysis and Applications, Kluwer Academic Publishers, Dordrecht–Boston–London, 2002 | MR

[50] Sviridyuk G.A., “The Manifold of Solutions of an Operator Singular Pseudoparabolic Equation”, Doklady Akademii Nauk SSSR, 289:6 (1986), 1–31 (in Russian) | MR

[51] Sviridyuk G.A., “On the Variety of Solutions of a Certain Problem of an Incompressible Viscoelastic Fluid”, Differential Equations, 24:10 (1988), 1846–1848 | MR

[52] Sviridyuk G.A., “A Problem for the Generalized Boussinesq Filtration Equation”, Soviet Mathematics, 33:2 (1989), 62–73 | MR | Zbl

[53] Sviridyuk G.A., “Solvability of the Viscoelastic Thermal Convection Problem of Incompressible Fluid”, Russian Mathematics, 1990, no. 12, 65–70 | MR | Zbl

[54] Sviridyuk G.A., “Sobolev-Type Linear Equations and Strongly Continuous Semigroups of Resolving Operators with Kernels”, Russian Academy of Sciences. Doklady. Mathematics, 50:1 (1995), 137–142 | MR

[55] Sviridyuk G.A., “Solvability of a Problem of the Thermoconvection of a Viscoelastic Incompressible Fluid”, Soviet Mathematics, 34:12 (1990), 80–86 | MR | Zbl

[56] Sviridyuk G.A., “On the General Theory of Operator Semigroups”, Russian Mathematical Surveys, 49:4 (1994), 45–74 | DOI | MR | Zbl

[57] Sviridyuk G.A., “Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type”, Russian Academy of Sciences. Izvestiya Mathematics, 42:3 (1994), 601–614 | DOI | MR

[58] Sviridyuk G.A., “About One Showalter Problem”, Differential Equations, 23:2 (1989), 338–339 | MR

[59] Sviridyuk G.A., Sukacheva T.G., “Phase Spaces of a Class of Operator Semilinear Equations of Sobolev Type”, Differential Equations, 26:2 (1990), 188–195 | MR | Zbl

[60] Sviridyuk G.A., Sukacheva T.G., “Cauchy Problem for a Class of Semilinear Equations of Sobolev Type”, Siberian Mathematical Journal, 31:5 (1990), 794–802 | DOI | MR | Zbl

[61] Sviridyuk G.A., Fedorov V.E., “Analytic Semigroups wit Kernels and Linear Equations of Sobolev Type”, Siberian Mathematical Journal, 36:5 (1995), 1130 | DOI | MR | Zbl

[62] Sviridyuk G.A., Efremov A.A., “An Optimal Control Problem for a Class of Linear Equations of Sobolev Type”, Russian Mathematics, 40:12 (1996), 60–71 | MR

[63] Sviridyuk G.A., Yakupov M.M., “The Phase Space of the Initial-Boundary Value Problem for the Oskolkov System”, Differential Equations, 32:11 (1996), 1535–1540 | MR | Zbl

[64] Sviridyuk G.A., Keller A.V., “Invariant Spaces and Dichotomies of Solutions of a Class of Linear Equations of Sobolev Type”, Russian Mathematics, 41:5 (1997), 57–65 | MR | Zbl

[65] Sviridyuk G.A., Efremov A.A., “Optimal Control for a Class of Degenerate Linear Equations”, Doklady Mathematics, 59:1 (1999), 157–159 | MR | Zbl

[66] Sviridyuk G.A., Kuznetsov G.A., “Relatively Strongly $p$-Sectorial Linear Operators”, Doklady Mathematics, 59:2 (1999), 298–300 | MR | Zbl

[67] Sviridyuk G.A., Zamyshlyaeva A.A., “Morphology of Phace Spaces of One Class of Linear Equations of Sobolev Type of High Order”, Bulletin of Chelyabinsk State University, 3:2(5) (1999), 999 | MR

[68] Sviridyuk G.A., Zagrebina S.A., “Verigin's Problem for Linear Equations of the Sobolev Type with Relatively $p$-Sectorial Operators”, Differential Equations, 38:12 (2002), 1745–1752 | DOI | MR | Zbl

[69] Sviridyuk G.A., Manakova N.A., “Regular Perturbations of a Class of Sobolev Type Linear Equations”, Differential Equations, 38:3 (2002), 447–450 | DOI | MR | Zbl

[70] Sviridyuk G.A., Kazak V.O., “The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation”, Mathematical Notes, 71:1–2 (2002), 262–266 | DOI | MR | Zbl

[71] Sviridyuk G.A., Brychev S.V., “Numerical Solution of Systems of Leontief Type Equations”, Russian Mathematics, 2003, no. 8, 46–52 | MR | Zbl

[72] Sviridyuk G.A., Burlachko I.V., “An Algorithm for Solving the Cauchy Problem for Degenerate Linear Systems of Ordinary Differential Equations”, Computational Mathematics and Mathematical Physics, 43:11 (2003), 1613–1619 | MR | Zbl

[73] Sviridyuk G.A., Shafranov D.E., “The Cauchy Problem for the Barenblatt–Zheltov–Kochina Equation on a Smooth Manifold”, Bulletin of Chelyabinsk State University, 9 (2003), 171–177 (in Russian) | MR

[74] Sviridyuk G.A., Ankudinov A.V., “The Phase Space of the Cauchy–Dirichlet Problem for a Nonclassical Equation”, Differential Equations, 39:11 (2003), 1639–1644 | DOI | MR | Zbl

[75] Sviridyuk G.A., Trineeva I.K., “A Whitney Fold in the Phase Space of the Hoff Equation”, Russian Mathematics, 49:10 (2005), 49–55 | MR | Zbl

[76] Sviridyuk G.A., Karamova A.F., “On the Crease Phase Space of One Non-Classical Equations”, Differential Equations, 41:10 (2005), 1476–1581 | DOI | MR

[77] Sviridyuk G.A., Manakova N.A., “The Phase Space of the Cauchy–Dirichlet Problem for the Oskolkov Equation of Nonlinear Filtration”, Russian Mathematics, 2003, no. 9, 33–38 | MR | Zbl

[78] Sviridyuk G.A., Kitaeva O.G., “Invariant Manifolds of the Hoff Equation”, Mathematical Notes, 79:3 (2006), 408–412 | DOI | MR | Zbl

[79] Sviridyuk G.A., Manakova N.A., “The Dynamical Models of Sobolv Type with Showalter–Sidorov Condition and Additive “Noise””, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 7:1 (2014), 90–103 | DOI | Zbl

[80] Sviridyuk G.A., Manakova N.A., “The Barenblatt–Zheltov–Kochina Model with Additive White Noise in Quasi-Sobolev Spaces”, Journal of Computational and Engineering Mathematics, 3:1 (2016), 61–67 | DOI | MR | Zbl

[81] Sviridyuk G.A., Shemetova V.V., “Hoff Equations on Graphs”, Differential Equations, 42:1 (2006), 139–145 | DOI | MR | Zbl

[82] Sviridyuk G.A., Fedorov V.E., Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston, 2003 | MR | Zbl

[83] Sviridyuk G.A., Zargebina S.A., “The Showalter–Sidorov Problem as a Phenomena of the Sobolev-Type Equations”, The Bulletin of Irkutsk State University. Series Mathematics, 3:1 (2010), 104–125 (in Russian) | MR | Zbl

[84] Sviridyuk G.A., Zamyshlyaeva A.A., “The Phase Spaces of a Class of Linear Higher-Order Sobolev Type Equations”, Differential Equations, 42:2 (2006), 269–278 | DOI | MR | Zbl

[85] Sviridyuk G.A., Zamyshlyaeva A.A., Zagrebina S.A., “Multipoint Initial-Final Problem for One Class of Sobolev Type Models of Higher Order with Additive White Noise”, Bulletin of the South Ural State University. Mathematical Modelling, Programming and Computer Software, 11:3 (2018), 103–117 | DOI | MR | Zbl

[86] Uvarova M.V., Pyatkov S.G., “Some Boundary Value Problems for the Sobolev-Type Operator-Differential Equations”, Mathematical Notes of SVFU, 26:3 (2019), 71–89 | MR

[87] Vragov V.N., Boundary value Problems for Non-Classical Mathematical Physics Equations, Novosibirsk State Univesity, Novosibirsk, 1983 | MR

[88] Zagrebina S.A., “The Initial-Finite Problems for Nonclassical Models of Mathematical Physics”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 6:2 (2013), 5–24 | MR | Zbl

[89] Zagrebina S. A., Konkina A. S., “Traffic Management Model”, Proceedings of 2nd International Conference on Industrial Engineering, Applications and Manufacturing, 2016, 7911712 | DOI

[90] Zagrebina S.A., Sagadeeva M.A., “The Generalized Splitting Theorem for Linear Sobolev type Equations in Relatively Radial Case”, The Bulletin of Irkutsk State University. Mathematics, 2014, no. 7, 19–33 | Zbl

[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, 113, 2015, 317–325 | DOI | MR | Zbl

[92] Zagrebina S., Sukacheva T., Sviridyuk G., “The Multipoint Initial-Final Value Problems for Linear Sobolev-Type Equations with Relatively $p$-Sectorial Operator and Additive “Noise””, Global and Stochastic Analysis, 5:2 (2018), 129–143

[93] Zamyshlyaeva A.A., “The Higher-Order Sobolev-Type Models”, Bulletin of the South Ural State University. Mathematical Modelling, Programming and Computer Software, 7:2 (2014), 5–28 | Zbl

[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, 8:4 (2015), 113–119 | DOI | Zbl

[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, 11:1 (2018), 5–14 | DOI | MR | Zbl

[96] Zamyshlyaeva A., Lut A., “Inverse Problem for the Sobolev Type Equation of Higher Order”, Mathematics, 9:14 (2021), 1647 | DOI | MR

[97] Zamyshlyaeva A. A., Manakova N. A., Tsyplenkova O. N., “Optimal Control in Linear Sobolev Type Mathematical Models”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 13:1 (2020), 5–27 | DOI | Zbl

[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, 8:4 (2016), 5–16 | DOI | MR | Zbl

[99] Zamyshlyaeva A.A., Tsyplenkova O.N., “The Optimal Control over Solutions of the Initial-finish Value Problem for the Boussinesque–Löve Equation”, Bulletin of the South Ural State University. Mathematical Modelling, Programming and Computer Software ,, 5(264) (2012), 13–24 (in Russian) | Zbl