Abstract
We consider some nonlinear evolution equation with an unbounded operator depending on a small parameter on the right-hand side and study the existence of solutions holomorphically depending on a parameter. We introduce the notion of \( \varepsilon \)-regular solution and establish the conditions for the \( \varepsilon \)-regular solution to coincide with a solution to this equation.
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References
Lomov S.A., Introduction to the General Theory of Singular Perturbations, Nauka, Moscow (1981) [Russian].
Lomov S.A. and Lomov I.S., Fundamentals of the Mathematical Theory of a Boundary Layer, Moscow University, Moscow (2011) [Russian].
Maslov V.P., Asymptotic Methods and Perturbation Theory, Nauka, Moscow (1988) [Russian].
Vasileva A.B. and Butuzov V.F., Asymptotic Decompositions of Solutions to the Singular Perturbation Problems, Nauka, Moscow (1973) [Russian].
Bibikov Yu.N., The General Course of Ordinary Differential Equations, Leningrad University, Leningrad (1981) [Russian].
Cercignani C., Mathematical Methods in Kinetic Theory, Springer, Boston (1990).
Krivoruchenko M.I., Nadyozhin D.K., and Yudin A.V., “Hydrostatic equilibrium of stars without electroneutrality constraint,” Phys. Rev. D., vol. 97, no. 15 (2018) (Article no. 083016).
Glizer V.Y., Fridman E., and Fedin Y., “A novel approach to exact slow-fast decomposition of linear singularly perturbed systems with small delays,” SIAM J. Control Optim., vol. 55, no. 1, 236–274 (2017).
Malek S., “On boundary layer expansions for a singularly perturbed problem with confluent Fuchsian singularities,” Mathematics, vol. 8(2), 189 (2020).
Glizer V.Y., “Asymptotic analysis of spectrum and stability for one class of singularly perturbed neutral-type time-delay systems,” Axioms, vol. 10, no. 4, 325 (2021).
Bobodzhanov A., Safonov V., and Kachalov V., “Asymptotic and pseudoholomorphic solutions of singularly perturbed differential and integral equations in the Lomov’s regularization method,” Axioms, vol. 8, no. 27 (2019) (27 pp.).
Reed M. and Simon B., Methods of Modern Mathematical Physics. Vol. 4: Analysis of Operators, Academic, New York (1972).
Richtmyer R., Principles of Advanced Mathematical Physics. Vol. 1, Springer, New York and Berlin (1978).
Krein S.G., Linear Differential Equations in Banach Space, Amer. Math. Soc., Providence (1971).
Daleckii Ju.L. and Krein M.G., Stability of Solutions to Differential Equations in Banach Space, Amer. Math. Soc., Providence (1974).
Trenogin V.A., Functional Analysis, Nauka, Moscow (1980) [Russian].
Ladyzhenskaya O.A., “Sixth problem of the millennium: Navier–Stokes equations, existence and smoothness,” Russian Math. Surveys, vol. 58, no. 2, 251–286 (2003).
Richtmyer R., Principles of Advanced Mathematical Physics. Vol. 2, Springer, New York and Berlin (1981).
Dezin A.A., Memories and Selected Works on Mathematics, Maks Press, Moscow (2011) [Russian].
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 1, pp. 113–122. https://doi.org/10.33048/smzh.2023.64.111
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Kachalov, V.I. On \( \varepsilon \)-Regular Solutions to Differential Equations with a Small Parameter. Sib Math J 64, 94–102 (2023). https://doi.org/10.1134/S0037446623010111
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DOI: https://doi.org/10.1134/S0037446623010111