Asymptotic Integration of Certain Differential Equations in Banach Space
https://doi.org/10.18255/1818-1015-2017-5-596-614
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.
References
1. Yosida K., Functional analysis, Springer-Verlag, Berlin, G¨ottingen, Heidelberg, 1965.
2. Coddington E. A., Levinson N., Theory of ordinary differential equations, McGraw-Hill, New York, 1955.
3. Marsden J. E., McCracken M., The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976.
4. Nesterov P. N., “Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients”, Differ. Equ., 43:6 (2007), 745–756.
5. Nesterov P. N., “Center manifold method in the asymptotic integration problem for functional differential equations with oscillatory decreasing coeffcients. II”, Model. Anal. Inform. Sist., 21:5 (2014), 5–37, (in Russian).
6. Fomin V. N., Mathematical theory of parametric resonance in linear distributed systems, Leningr. Univ. Publ., Leningrad, 1972, (in Russian).
7. Hale J. K., Theory of functional differential equations, Springer-Verlag, New York, 1977.
8. Balakrishnan A. V., Applied functional analysis, Springer-Verlag, New York, 1981.
9. Ball J. M., “Strongly continuous semigroups, weak solutions, and the variation of constants formula”, Proc. Amer. Math. Soc., 63:2 (1977), 370–373.
10. Ball J. M., “On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations”, J. Differential Equations, 27 (1978), 224–265.
11. Carr J., Applications of centre manifold theory, Springer-Verlag, New York, 1981.
12. Eastham M. S. P., The asymptotic solution of linear differential systems, Clarendon Press, Oxford, 1989.
13. Hale J., Verduyn Lunel S. M., Introduction to functional differential equations, SpringerVerlag, New York, 1993.
14. Kato T., Perturbation theory for linear operators, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
15. Langer M., Kozlov V., “Asymptotics of solutions of a perturbed heat equation”, J. Math. Anal. Appl., 397:2 (2013), 481–493.
16. Levinson N., “The asymptotic nature of solutions of linear systems of differential equations”, Duke Math. J., 15:1 (1948), 111–126
17. Nesterov P., “Asymptotic integration of functional differential systems with oscillatory decreasing coefficients: a center manifold approach”, Electron. J. Qual. Theory Differ. Equ., 2016, № 33, 1–43.
18. Pazy A., Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
19. Rothe F., Global solutions of reaction-diffusion systems, Springer-Verlag, Berlin, Heidelberg, 1984.
20. Wu J., Theory and applications of partial functional differential equations, Springer-Verlag, New York, 1996.
Review
For citations:
Nesterov P.N. Asymptotic Integration of Certain Differential Equations in Banach Space. Modeling and Analysis of Information Systems. 2017;24(5):596-614. (In Russ.) https://doi.org/10.18255/1818-1015-2017-5-596-614
Views:
858
Email this article 