Abstract:
The article surveys the results concerning the morphology of phase spaces for semilinear Sobolev type equations. The first three paragraphs present specific boundary value problems for Sobolev type partial differential equations whose phase spaces are simple smooth Banach manifolds. The last section contains the mathematical models whose phase spaces lie on a smooth Banach manifolds with singularities. The purpose of this article is the formation of a basis for future studies of the morphology of phase spaces for semilinear Sobolev type equations. In addition, the article provides an explanation of the phenomenon of nonexistence of solutions to the Cauchy problem and the phenomenon of nonuniqueness of solutions to the Showalter–Sidorov problem for the semilinear Sobolev type equations.
Keywords:
Sobolev type equations, phase space, the morphology of the phase space, Banach manifold, quasistationary trajectory, Showalter–Sidorov problem, Cauchy problem, $k$-assembly Whitney.
Received: 15.06.2016
Bibliographic databases:
Document Type:
Article
UDC:
517.9
Language: Russian
Citation:
N. A. Manakova, G. A. Sviridyuk, “Nonclassical equations of mathematical physics. Phase space of semilinear Sobolev type equations”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 8:3 (2016), 31–51
\Bibitem{ManSvi16}
\by N.~A.~Manakova, G.~A.~Sviridyuk
\paper Nonclassical equations of mathematical physics. Phase space of semilinear Sobolev type equations
\jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz.
\yr 2016
\vol 8
\issue 3
\pages 31--51
\mathnet{http://mi.mathnet.ru/vyurm307}
\crossref{https://doi.org/10.14529/mmph160304}
\elib{https://elibrary.ru/item.asp?id=26367651}
Linking options:
https://www.mathnet.ru/eng/vyurm307
https://www.mathnet.ru/eng/vyurm/v8/i3/p31
This publication is cited in the following 18 articles:
N. G. Nikolaeva, O. V. Gavrilova, N. A. Manakova, “Investigation of the uniqueness solution of the Showalter–Sidorov problem for the mathematical Hoff model. Phase space morphology”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 17:1 (2024), 49–63
A. V. Buevich, M. A. Sagadeeva, S. A. Zagrebina, “Stability of a stationary solution to one class of non-autonomous Sobolev type equations”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 16:3 (2023), 65–73
A. V. Keller, “O napravleniyakh issledovanii uravnenii sobolevskogo tipa”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 16:4 (2023), 5–32
N. A. Manakova, O. V. Gavrilova, K. V. Perevozchikova, “Polulineinye modeli sobolevskogo tipa. Needinstvennost resheniya zadachi Shouoltera – Sidorova”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 15:1 (2022), 84–100
K. V. Perevozhikova, N. A. Manakova, “Research of the optimal control problem for one mathematical model of the Sobolev type”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 14:4 (2021), 36–45
O. V. Gavrilova, “Chislennoe issledovanie odnoznachnoi razreshimosti zadachi Shouoltera – Sidorova dlya matematicheskoi modeli rasprostraneniya nervnykh impulsov v membranoi obolochke”, J. Comp. Eng. Math., 8:3 (2021), 32–48
K V Vasiuchkova, “Numerical study of the optimal control problem for one model of potential distribution in a crystalline semiconductor with the Showalter–Sidorov condition”, J. Phys.: Conf. Ser., 1847:1 (2021), 012024
K. V. Perevozhikova, N. A. Manakova, A. S. Kuptsova, “Issledovanie razlichnykh tipov zadach upravleniya dlya odnoi modeli nelineinoi filtratsii”, J. Comp. Eng. Math., 8:4 (2021), 45–61
Ksenia V. Vasiuchkova, Natalia A. Manakova, Georgy A. Sviridyuk, Springer Proceedings in Mathematics & Statistics, 325, Semigroups of Operators – Theory and Applications, 2020, 363
N. A. Manakova, K. V. Vasyuchkova, “Issledovanie odnoi matematicheskoi modeli raspredeleniya potentsialov v kristallicheskom poluprovodnike”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 12:2 (2019), 150–157
K. V. Vasiuchkova, “Chislennoe issledovanie dlya zadachi startovogo upravleniya i finalnogo nablyudeniya v modeli raspredeleniya potentsialov v kristallicheskom poluprovodnike”, J. Comp. Eng. Math., 6:3 (2019), 54–68
O. V. Gavrilova, “Chislennoe issledovanie needinstvennosti resheniya zadachi Shouoltera – Sidorova dlya odnoi vyrozhdennoi matematicheskoi modeli avtokataliticheskoi reaktsii s diffuziei”, J. Comp. Eng. Math., 6:4 (2019), 3–17
O. V. Gavrilova, “Numerical study of a mathematical model of an autocatalytic reaction with diffusion in a tubular reactor”, J. Comp. Eng. Math., 5:3 (2018), 24–37
N. A. Manakova, O. V. Gavrilova, “About nonuniqueness of solutions of the Showalter–Sidorov problem for one mathematical model of nerve impulse spread in membrane”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 11:4 (2018), 161–168
P. O. Moskvicheva, “A numerical experiment for the Barenblatt – Zheltov – Kochina equation in a bounded domain”, J. Comp. Eng. Math., 4:2 (2017), 41–48
E. V. Bychkov, K. Yu. Kotlovanov, “Sobolev type equation in $(n, p)$-sectorial case”, J. Comp. Eng. Math., 4:2 (2017), 66–72
A. A. Zamyshlyaeva, G. A. Sviridyuk, “Nonclassical equations of mathematical physics. Linear Sobolev type equations of higher order”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 8:4 (2016), 5–16
N. A. Manakova, “On modified method of multistep coordinate descent for optimal control problem for semilinear Sobolev-type model”, J. Comp. Eng. Math., 3:4 (2016), 59–72
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