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 Vestnik YuUrGU. Ser. Mat. Model. Progr., 2018, Volume 11, Issue 4, Pages 161–168 (Mi vyuru465)

Short Notes

About nonuniqueness of solutions of the Showalter–Sidorov problem for one mathematical model of nerve impulse spread in membrane

N. A. Manakova, O. V. Gavrilova

South Ural State University, Chelyabinsk, Russian Federation

Abstract: The article is devoted to the study of the morphology of the phase space of a mathematical model of the nerve impulse spread in a membrane, based on a degenerate Fitz Hugh–Nagumo system, defined on a bounded domain with a smooth boundary. In this mathematical model, the rate of change of one of the components of the system can significantly exceed the other, which leads to a degenerate Fitz Hugh–Nagumo system. The model under inquiry belongs to a wide class of semilinear Sobolev type models. To research the problem of nonuniqueness of solutions of the Showalter–Sidorov problem, the phase space method will be used, which was developed by G.A. Sviridyuk to scrutinize the solvability of Sobolev type equations. We have shown that the phase space of the studied model contains singularity such as the Whitney fold. The conditions of existence, uniqueness or multiplicity of solutions of the Showalter–Sidorov problem depending on the parameters of the system are found.

Keywords: Sobolev type equations, Showalter–Sidorov problem, Fitz Hugh–Nagumo system, nonuniqueness of the solution.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 02.A03.21.0011 The work was supported by Act 211 Government of the Russian Federation, contract no. 02.A03.21.0011.

DOI: https://doi.org/10.14529/mmp180413

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UDC: 517.9
MSC: 60H30
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Citation: 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”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 11:4 (2018), 161–168

Citation in format AMSBIB
\Bibitem{ManGav18} \by N.~A.~Manakova, O.~V.~Gavrilova \paper About nonuniqueness of solutions of the Showalter--Sidorov problem for one mathematical model of nerve impulse spread in membrane \jour Vestnik YuUrGU. Ser. Mat. Model. Progr. \yr 2018 \vol 11 \issue 4 \pages 161--168 \mathnet{http://mi.mathnet.ru/vyuru465} \crossref{https://doi.org/10.14529/mmp180413} \elib{http://elibrary.ru/item.asp?id=36487062}