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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2020, Volume 13, Issue 2, Pages 17–32 (Mi vyuru540)  

Mathematical Modelling

Positive solutions to Sobolev type equations with relatively $p$-sectorial operators

J. Banasiakab, N. A. Manakovaa, G. A. Sviridyuka

a South Ural State University, Chelyabinsk, Russian Federation
b University of Pretoria, Pretoria, South Africa

Abstract: The article describes sufficient conditions for the existence of positive solutions to both the Cauchy problem and the Showalter–Sidorov problem for an abstract linear Sobolev type equation. A distinctive feature of such equations is the phenomenon of non-existence and non-uniqueness of solutions. The research is based on the theory of positive semigroups of operators and the theory of degenerate holomorphic semigroups of operators. The merger of these theories leads to a new theory of degenerate positive holomorphic semigroups of operators. In spaces of sequences, which are analogues of Sobolev function spaces, the constructed abstract theory is used to study a mathematical model. The results can be used to study economic and engineering problems.

Keywords: Sobolev type equations, positive degenerate holomorphic semigroups of operators, positive solution, Sobolev sequence spaces.

DOI: https://doi.org/10.14529/mmp200202

Full text: PDF file (262 kB)
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UDC: 517.9
MSC: 43A35, 47B37
Received: 21.05.2019
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Citation: J. Banasiak, N. A. Manakova, G. A. Sviridyuk, “Positive solutions to Sobolev type equations with relatively $p$-sectorial operators”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 13:2 (2020), 17–32

Citation in format AMSBIB
\Bibitem{BanManSvi20}
\by J.~Banasiak, N.~A.~Manakova, G.~A.~Sviridyuk
\paper Positive solutions to Sobolev type equations with relatively $p$-sectorial operators
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2020
\vol 13
\issue 2
\pages 17--32
\mathnet{http://mi.mathnet.ru/vyuru540}
\crossref{https://doi.org/10.14529/mmp200202}


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