{"title":"具有无界算子的进化方程的全局可解性","authors":"A. Chernov","doi":"10.35634/vm210212","DOIUrl":null,"url":null,"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\n\\[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,\\]\nwhere $u=u(t)\\colon[0;T]\\to U$ 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.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":"112 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On totally global solvability of evolutionary equation with unbounded operator\",\"authors\":\"A. Chernov\",\"doi\":\"10.35634/vm210212\",\"DOIUrl\":null,\"url\":null,\"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\\n\\\\[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,\\\\]\\nwhere $u=u(t)\\\\colon[0;T]\\\\to U$ 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.\",\"PeriodicalId\":43239,\"journal\":{\"name\":\"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki\",\"volume\":\"112 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2021-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.35634/vm210212\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.35634/vm210212","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On totally global solvability of evolutionary equation with unbounded operator
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.