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## Archive of Issues

Russia Nizhni Novgorod
Year
2021
Volume
31
Issue
2
Pages
331-349
 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. 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