Полулинейные модели соболевского типа. Неединственность решения задачи Шоуолтера - Сидорова

Автор: Манакова Наталья Александровна, Гаврилова Ольга Витальевна, Перевозчикова Ксения Владимировна

Журнал: Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование @vestnik-susu-mmp

Рубрика: Обзорные статьи

Статья в выпуске: 1 т.15, 2022 года.

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Статья имеет обзорный характер и содержит результаты исследования морфологии фазовых пространств полулинейных моделей соболевского типа. Представлены исследования тех математических моделей, чьи фазовые пространства лежат на гладких банаховых многообразиях с особенностями в зависимости от параметров задачи, а именно, модели Хоффа, модели Плотникова, модели распределенного брюсселятора и модели распространения нервного импульса. В первой части статьи приведены условия, при которых фазовые многообразия изучаемых моделей - простые гладкие банаховы многообразия, из чего вытекает единственность решения задачи Шоуолтера - Сидорова. Во второй части статьи приведены условия, при которых фазовые многообразия исследуемых моделей содержат особенности, из чего вытекает неединственность решения задачи Шоуолтера - Сидорова.

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Уравнения соболевского типа, фазовое пространство, морфология фазового пространства, банаховы многообразия, задача шоуолтера - сидорова, k-сборка уитни

Короткий адрес: https://sciup.org/147237431

IDR: 147237431

Список литературы Полулинейные модели соболевского типа. Неединственность решения задачи Шоуолтера - Сидорова

Al’shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-Up in Nonlinear Sobolev Type Equations.Berlin, Walter de Gruyter, 2011. DOI: 10.1515/9783110255294
Borisov, V.G. Parabolic Boundary Value Problems with a Small Parameter Multiplying the Derivatives with Respect to t. Mathematics of the USSR-Sbornik, 1988, vol. 59, no. 2, pp. 287–302.
Bokareva T.A., Sviridyuk G.A. Whitney Folds in Phase Spaces of Some Semilinear Sobolev-Type Equations. Mathematical Notes, 1994, vol. 55, no. 3, pp. 237–242. DOI: 10.1007/BF02110776
Favini A., Sviridyuk G., Sagadeeva M. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of “Noises”. Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4607–4621. DOI: 10.1007/s00009-016-0765-x
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, vol. 2018, article ID: 128, 10 p.
Field R.J., Koros E., Noyes R. Oscillations in Chemical Systems. II. Thorough Analysis of Temporal Oscillation in the Bromate-Cerium-Malonic Acid System. Journal of the American Chemical Society, 1972, vol. 94, no. 25, pp. 8649–8664.
Fisher R.A. The Wave of Advance of Advantageous Genes. Annals of Human Genetics, 1937, no. 7, pp. 353–369.
Fitzhugh R. Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal, 1961, vol. 1, no. 6, pp. 445–466.
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, 2019, vol. 6, no. 4, pp. 3–17. DOI: 10.14529/jcem190401
Gavrilova O.V., Nikolaeva N.G., Manakova N.A. [On the Non-Uniqueness of Solutions of the Showalter–Sidorov Problem for One Mathematical Model of the Deformation of an I-Beam]. South Ural Youth School of Mathematical Modelling, 2021, pp. 67–71. (in Russian)
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, pp. 85–90. (in Russian)
Hoff N.J. Creep Buckling. Journal of Aeronautical Sciences, 1956, no. 7, pp. 1–20.
Кeller A.V. Leontief Type Systems: Classes of Problems with Showalter–Sidorov Initial Condition and Numerical Solutions. The Bulletin of Irkutsk State University. Series: Mathematics, 2010, vol. 3, no. 2, pp. 30–43. (in Russian)
Kondyukov A.O., Sukacheva T.G. Phase Space of the Initial-Boundary Value Problem for the Oskolkov System of Nonzero Order. Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 5, pp. 823–829. DOI: 10.1134/S0965542515050127
Kondyukov A.O., Sukacheva T.G. Phase Space of a Model of Magnetohydrodynamics. Differential Equations, 2015, vol. 51, no. 4, pp. 502–509.
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, 2021, vol. 85, no. 1, pp. 111–144. DOI: 10.4213/im8949
Kozhanov A.I. Boundary Value Problems for Fourth-Order Sobolev Type Equations. Journal of Siberian Federal University. Mathematics and Physics, 2021, vol. 14, no. 4, pp. 425–432.
Lefever R. Symmetry-Breaking Instabilities in Dissipative System. Journal of American Physics, 1968, no. 48, pp. 1695–1700.
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, 2016, vol. 8, no. 3, pp. 31–51. DOI: 10.14529/mmph160304
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, 2018, vol. 11, no. 4, pp. 161–168. DOI:10.14529/mmp180413
Nagumo J., Arimoto S., Yoshizawa S. An Active Pulse Transmission Line Simulating Nerve Axon. IRE Transactions on Nuclear Science, 1962, no. 50, no. 10, pp. 2061–2071.
Plotnikov P.I., Starovoitov V.N. The Stefan Problem with Surface Tension as a Limit of the Phase Field Model. Differential Equations, 1993, vol. 29, no. 3, pp. 395–405.
Plotnikov P.I., Klepacheva A.V. The Phase Field Equations and Gradient Flows of Marginal Functions. Siberian Mathematical Journal, 2001, vol. 42, no. 3, pp.551–567.
Pyatkov S.G. Boundary Value and Inverse Problems for Some Classes of Nonclassical Operator-Differential Equations. Siberian Mathematical Journal, 2021, vol. 62, no. 3, pp. 489–502. DOI: 10.1134/S0037446621030125
Sviridyuk G.A. The Manifold of Solutions of an Operator Singular Pseudoparabolic Equation. Doklady Akademii Nauk SSSR, 1986, vol. 289, no. 6, pp. 1–31. (in Russian)
Sviridyuk G.A. A Problem for the Generalized Boussinesq Filtration Equation. Soviet Mathematics (Izvestiya VUZ. Matematika), 1989, vol. 33, no. 2, pp. 62–73.
Sviridyuk G.A. About One Showalter Problem. Differential Equations, 1989, vol. 23, no. 2, pp. 338–339.
Sviridyuk G.A., Sukacheva T.G. Phase Spaces of a Class of Operator Semilinear Equations of Sobolev Type. Differential Equations, 1990, vol. 26, no. 2, pp. 188–195.
Sviridyuk G.A. Solvability of the Viscoelastic Thermal Convection Problem of Incompressible Fluid. Russian Mathematics, 1990, no. 12, pp. 65–70.
Sviridyuk G.A. Sobolev-Type Linear Equations and Strongly Continuous Semigroups of Resolving Operators with Kernels. Russian Academy of Sciences. Doklady. Mathematics, 1995, vol. 50, no. 1, pp. 137–142.
Sviridyuk G.A., Efremov A.A. Optimal Control for a Class of Degenerate Linear Equations. Doklady Mathematics, 1999, vol. 59, no. 1, pp. 157–159.
Sviridyuk G.A., Kazak V.O. The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation. Mathematical Notes, 2002, vol. 71, no. 1–2, pp. 262–266. DOI: 10.1023/A:1013919500605
Sviridyuk G.A., Trineeva I.K. A Whitney Fold in the Phase Space of the Hoff Equation. Russian Mathematics (Izvestiya VUZ. Matematika), 2005, vol. 49, no. 10, pp. 49–55.
Sviridyuk G.A., Karamova A.F. On the Crease Phase Space of One Non-Classical Equations. Differential Equations, 2005, vol. 41, no. 10, pp. 1476–1581. DOI: 10.1007/s10625-005-0300-5
Sviridyuk G.A., Shemetova V.V. Hoff Equations on Graphs. Differential Equations, 2006, vol. 42, no. 1, pp. 139–145. DOI: 10.1134/S0012266106010125.
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, 2010, vol. 3, no. 1, pp. 104–125. (in Russian)
Thompson J.M.T. Instabilities and Catastrophes in Sciences and Engineering. Chichester, New York, Brisbane, Toronto, Singapore, John Wiley and Sons, 1982.
Uvarova M.V., Pyatkov S.G. Some Boundary Value Problems for the Sobolev-Type Operator- Differential Equations. Mathematical Notes of SVFU, 2019, vol. 26, no. 3, pp. 71–89.
Zagrebina S.A., Konkina A.S. The Multipoint Initial-Final Value Condition for the Navier– Stokes Linear Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 1, pp. 132–136. DOI: 10.14529/mmp150111
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
Zamyshlyaeva A.A., Bychkov E.V., Tsyplenkova O.N. Mathematical Models Based on Boussinesq–Love Equation. Applied Mathematical Sciences, 2014, vol. 8, no. 110, pp. 5477–5483.
Zamyshlyaeva A.A., Keller A.V., Syropiatov M.B. Stochastic Model of Optimal Dynamic Measurements. Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 2, pp. 147–153. DOI: 10.14529/mmp180212

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