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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, Volume 31, Issue 2, Pages 331–349
DOI: https://doi.org/10.35634/vm210212
(Mi vuu773)
 

This article is cited in 2 scientific papers (total in 2 papers)

MATHEMATICS

On totally global solvability of evolutionary equation with unbounded operator

A. V. Chernovab

a Nizhni Novgorod State University, pr. Gagarina, 23, Nizhny Novgorod, 603950, Russia
b Nizhni Novgorod State Technical University, ul. Minina, 24, Nizhny Novgorod, 603950, Russia
Full-text PDF (277 kB) Citations (2)
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Abstract: Let XX be a Hilbert space, UU be a Banach space, G:XXG:XX be a linear operator such that the operator Bλ=λIGBλ=λIG is maximal monotone with some (arbitrary given) λR. For the Cauchy problem associated with controlled semilinear evolutionary equation as follows
x(t)=Gx(t)+f(t,x(t),u(t)),t[0;T];x(0)=x0X,
where u=u(t):[0;T]U is a control, x(t) is unknown function with values in X, we prove the totally (with respect to a set of admissible controls) global solvability subject to global solvability of the Cauchy problem associated with some ordinary differential equation in the space R. Solution x is treated in weak sense and is sought in the space Cw([0;T];X) of weakly continuous functions. In fact, we generalize a similar result having been proved by the author formerly for the case of bounded operator G. The essence of this generalization consists in that postulated properties of the operator Bλ give us the possibility to construct Yosida approximations for it by bounded linear operators and thus to extend required estimates from “bounded” to “unbounded” case. As examples, we consider initial boundary value problems associated with the heat equation and the wave equation.
Keywords: semilinear evolutionary equation in a Hilbert space, maximal monotone operator, totally global solvability.
Received: 28.08.2020
Bibliographic databases:
Document Type: Article
UDC: 517.957, 517.988, 517.977.56
MSC: 47J05, 47J35, 47N10
Language: Russian
Citation: A. V. Chernov, “On totally global solvability of evolutionary equation with unbounded operator”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 31:2 (2021), 331–349
Citation in format AMSBIB
\Bibitem{Che21}
\by A.~V.~Chernov
\paper On totally global solvability of evolutionary equation with unbounded operator
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2021
\vol 31
\issue 2
\pages 331--349
\mathnet{http://mi.mathnet.ru/vuu773}
\crossref{https://doi.org/10.35634/vm210212}
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  • https://www.mathnet.ru/eng/vuu/v31/i2/p331
  • This publication is cited in the following 2 articles:
    1. A. V. Chernov, “On the Exact Controllability of a Semilinear Evolution Equation with an Unbounded Operator”, Diff Equat, 59:2 (2023), 265  crossref
    2. A. V. Chernov, “O totalno globalnoi razreshimosti evolyutsionnogo uravneniya s monotonnym nelineinym operatorom”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:1 (2022), 130–149  mathnet  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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