Abstract
In this work, we investigate the problem of constructing asymptotic representations for weak solutions of a certain class of linear differential equations in the Banach space as an independent variable tends to infinity. We consider the class of equations that represent a perturbation of a linear autonomous equation, in general, with an unbounded operator. The perturbation takes the form of a family of bounded operators that, in a sense, oscillatorally decreases at infinity. It is assumed that 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 of a center-like manifold (a critical manifold) for the initial equation. This manifold is positively invariant with respect to the initial equation and attracts all trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional system of ordinary differential equations. The asymptotics of the fundamental matrix of this system can be constructed by using the method developed by P.N. Nesterov for asymptotic integration of systems with oscillatory decreasing coefficients. We illustrate the proposed technique by constructing the asymptotic representations for solutions of the perturbed heat equation.
Similar content being viewed by others
REFERENCES
Yoshida, L., Functional Analysis, Berlin–Gottingen–Heidelberg: Springer-Verlag, 1965.
Coddington, E.A. and Levinson, N., Theory of Ordinary Differential Equations, New York: McGraw-Hill, 1955.
Marsden, J.E. and McCracken, M., The Hopf Bifurcation and Its Applications, New York: Springer-Verlag, 1976.
Nesterov, P.N., Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients, Differ. Equations, 2007, vol. 43, no. 6, pp. 745–756.
Nesterov, P.N., Center manifold method in the asymptotic integration problem for functional differential equations with oscillatory decreasing coefficients. II, Model. Anal. Inf. Sist., 2014, vol. 21, no. 5, pp. 5–37.
Fomin, V.N., Mathematical Theory of Parametric Resonance in Linear Distributed Systems, Leningrad: Leningr. Gos. Univ., 1972.
Hale, J.K., Theory of Functional Differential Equations, New York: Springer-Verlag, 1977.
Balakrishnan, A.V., Applied Functional Analysis, New York: Springer-Verlag, 1981.
Ball, J.M., Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Am. Math. Soc., 1977, vol. 63, no. 2, pp. 370–373.
Ball, J.M., On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, J. Differ. Equations, 1978, vol. 27, pp. 224–265.
Carr, J., Applications of Centre Manifold Theory, New York: Springer-Verlag, 1981.
Eastham, M.S.P., The Asymptotic Solution of Linear Differential Systems, Oxford: Clarendon Press, 1989.
Hale, J. and Verduyn Lunel, S.M., Introduction to Functional Differential Equations, New York: Springer-Verlag, 1993.
Kato, T., Perturbation Theory for Linear Operators, Berlin–Heidelberg–New York: Springer-Verlag, 1980.
Langer, M. and Kozlov, V., Asymptotics of solutions of a perturbed heat equation, J. Math. Anal. Appl., 2013, vol. 397, no. 2, pp. 481–493.
Levinson, N., The asymptotic nature of solutions of linear systems of differential equations, Duke Math. J., 1948, vol. 15, no. 1, pp. 111–126.
Nesterov, P., Asymptotic integration of functional differential systems with oscillatory decreasing coefficients: A center manifold approach, Electron. J. Qual. Theory Differ. Equations, 2016, no. 33, pp. 1–43.
Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer-Verlag, 1983.
Rothe, F., Global Solutions of Reaction-Diffusion Systems, Berlin–Heidelberg: Springer-Verlag, 1984.
Wu, J., Theory and Applications of Partial Functional Differential Equations, New York: Springer-Verlag, 1996.
Funding
The reported study was funded by RFBR according to the research project 18-29-10055.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
Translated by L. Kartvelishvili
About this article
Cite this article
Nesterov, P.N. Asymptotic Integration of Certain Differential Equations in Banach Space. Aut. Control Comp. Sci. 53, 755–768 (2019). https://doi.org/10.3103/S0146411619070150
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0146411619070150