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Model. Anal. Inform. Sist., 2017, Volume 24, Number 5, Pages 596–614 (Mi mais586)  

Asymptotic integration of certain differential equations in Banach space

P. N. Nesterov

P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl, 150003, Russia

Abstract: We investigate the problem of constructing the asymptotics for weak solutions of certain class of linear differential equations in the Banach space as the independent variable tends to infinity. The studied class of equations is the perturbation of linear autonomous equation, generally speaking, with an unbounded operator. The perturbation takes the form of the family of the bounded operators that, in a sense, decreases oscillatory at infinity. The unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence for the initial equation of the center-like manifold (critical manifold). This manifold is positively invariant with respect to the initial equation and attracts all the trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional ordinary differential system. The asymptotics for the fundamental matrix of this system may be constructed by using the method proposed by the author for asymptotic integration of the systems with oscillatory decreasing coefficients. We illustrate the suggested technique by constructing the asymptotic formulas for solutions of the perturbed heat equation.

Keywords: asymptotics, differential equation, Banach space, oscillatory decreasing coefficients, center manifolds method, perturbed heat equation.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation МК-4625.2016.1
This research was supported by the grant of the President of the Russian Federation No. MK-4625.2016.1.


DOI: https://doi.org/10.18255/1818-1015-2017-5-596-614

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UDC: 517.968.72
Received: 26.03.2017

Citation: P. N. Nesterov, “Asymptotic integration of certain differential equations in Banach space”, Model. Anal. Inform. Sist., 24:5 (2017), 596–614

Citation in format AMSBIB
\Bibitem{Nes17}
\by P.~N.~Nesterov
\paper Asymptotic integration of certain differential equations in Banach space
\jour Model. Anal. Inform. Sist.
\yr 2017
\vol 24
\issue 5
\pages 596--614
\mathnet{http://mi.mathnet.ru/mais586}
\crossref{https://doi.org/10.18255/1818-1015-2017-5-596-614}
\elib{http://elibrary.ru/item.asp?id=30353170}


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