Geodesic
Positive solutions to Sobolev type equations with relatively $p$-sectorial operators
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 13 (2020) no. 2, pp. 17-32

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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: positive degenerate holomorphic semigroups of operators, Sobolev sequence spaces.
Mots-clés : Sobolev type equations, positive solution 
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     title = {Positive solutions to {Sobolev} type equations with relatively $p$-sectorial operators},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
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J. Banasiak; N. A. Manakova; G. A. Sviridyuk. Positive solutions to Sobolev type equations with relatively $p$-sectorial operators. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 13 (2020) no. 2, pp. 17-32. http://geodesic.mathdoc.fr/item/VYURU_2020_13_2_a1/