# A singularly perturbed Cauchy problem for a parabolic equation in the presence of the “weak” turning point of the limit operator

### Abstract

The article is devoted to the development of S. A. Lomov’s regularization method for singularly perturbed Cauchy problems in the case of violation of the stability conditions for the spectrum of the limit operator. In particular, the problem is considered in the presence of a “weak” turning point, in which the eigenvalues “stick together” at the initial moment of time. Problems with this kind of spectral features are well known to specialists in mathematical and theoretical physics, as well as in the theory of differential equations; however, they have not been previously considered from the point of view of the regularization method. Our work fills this gap. Based on the ideas of S. A. Lomov and A. G. Eliseev for asymptotic integration of problems with spectral features, it indicates how to introduce regularizing functions, describes in detail the algorithm of the regularization method in the case of a “weak” turning point, and justifies this algorithm; an asymptotic solution of any order with respect to a small parameter is constructed.

### References

[1]

Vishik M. I. and Lyusternik L. A., “Regular degeneration and boundary layer for linear differential equations with a small parameter,” Russ. Math. Surv., 12, No. 5, 3–122 (1957).

[2]

Langer R. E., “The asymptotic solutions of ordinary linear differential equations of the second order with special reference to a turning point,” Trans. Amer. Math. Soc., 67, 461–490 (1949).

[3]

Trenogin V. A., “Development and applications of the Lyusternik–Vishik asymptotic method,” Russ. Math. Surv., 25, No. 4, 123–156 (1970).

[4]

Bogolyubov N. N. and Mitropolsky Yu. A., Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1963).

[5]

Vasilyeva A. B. and Butuzov V. F., Asymptotic Expansions of Solutions of Singularly Perturbed Equations [in Russian], Nauka, Moscow (1973).

[6]

Maslov V. P., The Complex Method of WKB in Nonlinear Equations [in Russian], Nauka, Moscow (1977).

[7]

Lomov S. A. and Eliseev A. G., “Asymptotic integration of singularly perturbed problems,” Russ. Math. Surv., 43, No. 3, 1–63 (1988).

[8]

Bobodzhanov A. A. and Safonov V. F., “Regularized asymptotics of solutions to integrodifferential partial differential equations with rapidly varying kernels,” Ufa Math. J., 10, No. 2, 3–13 (2018).

[9]

Kucherenko V. V., “Asymptotics of solutions of the system $A(x,-ih)\frac{\partial}{\partial x}$ as h → 0 in the case of characteristics of variable multiplicity,” Math. USSR Izv., 8, No. 3, 631–666 (1974).

[10]

Kucherenko V. V. and Osipov Yu. V., “Exact and asymptotic solutions of systems with turning points,” Math. USSR-Izv., 29, No. 2, 355–370 (1987).

[11]

Lomov S. A. and Safonov V. F., “Regularization and asymptotic solutions of singularly perturbed problems with point features of the limit operator spectrum [in Russian],” Ukr. Math. J., 36, No. 2, 172–180 (1984).

[12]

Eliseev A. G. and Lomov S. A., “The theory of singular perturbations in the case of spectral singularities of a limit operator,” Math. USSR Sb., 59, No. 2, 541–555 (1988).

[13]

Eliseev A. G. and Salnikova T. A., “Construction of a solution to the Cauchy problem in the case of a weak turning point of the limit operator [in Russian],” in: Matematicheskie Metody i Prilozheniya, Mater. Mat. Chtenii RGSU, pp. 46–52, RGSU, Ruza (2011).

[14]

Lomov S. A., Introduction to the General Theory of Singular Perturbations [in Russian], Nauka, Moscow (1981).

*Mathematical notes of NEFU*, 27(3), pp. 3-15. doi: https://doi.org/10.25587/SVFU.2020.43.25.001.

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