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Sbornik: Mathematics, 2020, Volume 211, Issue 5, Pages 643–688
DOI: https://doi.org/10.1070/SM9261 (Mi sm9261) 		 
The Cauchy problem for an abstract second order ordinary differential equation

V. S. Gavrilov

National Research Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
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DOI: https://doi.org/10.1070/SM9261
Abstract: We prove the existence and uniqueness of a solution for the Cauchy problem for a linear abstract second order differential equation, obtain its representation, and prove that it is continuously dependent on the time at which the initial conditions are specified. Based on these results, we prove the existence and uniqueness of a solution of the Cauchy problem for a nonlinear abstract second order differential equation. This result is applied to show that the initial-boundary value problem for a nonlinear hyperbolic divergence structure equation has a unique solution.
Bibliography: 49 titles.
Keywords: hyperbolic equation, partial differential equation, abstract equation.
Received: 12.04.2019
Russian version:
Matematicheskii Sbornik, 2020, Volume 211, Number 5, Pages 31–77
DOI: https://doi.org/10.4213/sm9261
Bibliographic databases:
Document Type: Article
UDC: 517.968.74
MSC: 45J05, 47G20, 34K30
Language: English
Original paper language: Russian
Citation: V. S. Gavrilov, “The Cauchy problem for an abstract second order ordinary differential equation”, Mat. Sb., 211:5 (2020), 31–77; Sb. Math., 211:5 (2020), 643–688
Citation in format AMSBIB
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    Abstract page:388
    Russian version PDF:38
    English version PDF:22
    References:45
    First page:35
     
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