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Ufa Mathematical Journal, 2021, Volume 13, Issue 4, Pages 63–79
DOI: https://doi.org/10.13108/2021-13-4-63
(Mi ufa592)
 

This article is cited in 3 scientific papers (total in 3 papers)

Exponential stability of semigroups generated by Volterra integro-differential equations

N. A. Rautian

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow Center of Fundamental and Applied Mathematics, GSP-1, Leninskie gory 1, 119991, Moscow, Russia
References:
Abstract: We study abstract Volterra integro-differential equations, which are operator models of problems in the viscoelasticity theory. This class includes Gurtin-Pipkin integro-differential equations describing the heat transfer in medias with memory. In particular, as the kernels of integral operators, the sums of decaying exponentials can serve or the sums of Rabotnov functions with positive coefficients having wide applications in the viscoelasticity theory and the theory of heat transfer.
The presented results are based on the approach related with studying one-parametric semi-groups for linear evolution equations. We provide a method for reducing the initial problem for a model integro-differential equation with operator coefficients in the Hilbert space to the Cauchy problem for a first order differential equation. We prove results on existing a strongly continuous contracting semigroup generated by a Volterra integro-differential equation with operator coefficients in a Hilbert space. We establish an exponential decay of the semigroup under known assumptions for the kernels of the integral operators. On the base of the obtained results we establish a well-posedness of initial problem for the Volterra integro-differential equation with appropriate estimates for the solution.
The proposed approach can be also employed for studying other integro-differential equations involving integral terms of Volterra convolution type.
Keywords: Volterra integro-differential equations, linear equations in Hilbert space, operator semigroups.
Funding agency Grant number
Russian Foundation for Basic Research 20-01-00288_а
The research is made in the framework of Program of Developing Interdisciplinary Scientific Educational School of Moscow University “Mathematical methods of analysis of complex system” under a financial support of Russian Foundation for Basic Researches (project no. 20-01-00288 A).
Received: 27.02.2021
Bibliographic databases:
Document Type: Article
UDC: 517.968.72
Language: English
Original paper language: Russian
Citation: N. A. Rautian, “Exponential stability of semigroups generated by Volterra integro-differential equations”, Ufa Math. J., 13:4 (2021), 63–79
Citation in format AMSBIB
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\by N.~A.~Rautian
\paper Exponential stability of semigroups generated by Volterra integro-differential equations
\jour Ufa Math. J.
\yr 2021
\vol 13
\issue 4
\pages 63--79
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\crossref{https://doi.org/10.13108/2021-13-4-63}
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  • https://doi.org/10.13108/2021-13-4-63
  • https://www.mathnet.ru/eng/ufa/v13/i4/p65
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Уфимский математический журнал
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    English version PDF:32
    References:37
     
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