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Home » Issues » Archive of Issues » 2021 - 2 issue » pp. 331-349

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Russia Nizhni Novgorod
Year
2021
Volume
31
Issue
2
Pages
331-349
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Section Mathematics
Title On totally global solvability of evolutionary equation with unbounded operator
Author(-s) Chernov A.V.ab
Affiliations Nizhni Novgorod State Technical Universitya, Nizhni Novgorod State Universityb
Abstract Let $X$ be a Hilbert space, $U$ be a Banach space, $G\colon X\to X$ be a linear operator such that the operator $B_\lambda=\lambda I-G$ is maximal monotone with some (arbitrary given) $\lambda\in\mathbb{R}$. For the Cauchy problem associated with controlled semilinear evolutionary equation as follows \[x^\prime(t)=Gx(t)+f\bigl( t,x(t),u(t)\bigr),\quad t\in[0;T];\quad x(0)=x_0\in X,\] where $u=u(t)\colon[0;T]\to U$ 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 $\mathbb{R}$. Solution $x$ is treated in weak sense and is sought in the space $\mathbb{C}_w\bigl([0;T];X\bigr)$ 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 $B_\lambda$ 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
UDC 517.957, 517.988, 517.977.56
MSC 47J05, 47J35, 47N10
DOI 10.35634/vm210212
Received 28 August 2020
Language Russian
Citation Chernov A.V. On totally global solvability of evolutionary equation with unbounded operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 331-349.
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