Skip to main content
Log in

Asymptotics of periodic solutions of autonomous parabolic equations with small diffusion

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Literature Cited

  1. A. B. Vasil'eva, S. V. Dvoryaninov, and N. Kh. Rozov, “Asymptotic theory of oscillations in singularly perturbed parabolic systems,” in: Ninth International Conference on Nonlinear Oscillations; Abstracts of Reports [in Russian], Naukova Dumka, Kiev (1981), p. 79.

    Google Scholar 

  2. A. B. Vasil'eva, “Application of the method of boundary functions to the construction of periodic solutions in problems with partial derivatives,” in: Methods of Small Parameters and Their Application [in Russian], IM AN BSSR, Minsk (1982), pp. 19–22.

    Google Scholar 

  3. Yu. S. Kolesov, “Problems of mathematical ecology,” in: Differential Equations and Their Application [in Russian], Inst. Mat. Kibern. Akad. Nauk LitSSR, Vilnius, Vol. 29, pp. 27–34.

  4. S. V. Dvoryaninov, “Periodic solution of an autonomous singularly perturbed parabolic system,” Differents. Uravn.,16, No. 9, 1617–1622 (1980).

    Google Scholar 

  5. S. A. Kaschenko, “Biological explanation of some laws of operation of the simplest ecosystems in extremal cases,” in: Research on Stability and the Theory of Oscillations [in Russian], Yaroslavl' State Univ. (1982), pp. 85–103.

  6. A. B. Vasil'eva, and V. F. Butuzov, Asymptotic Expansion of Singularly Perturbed Equations [in Russian], Nauka, Moscow (1973).

    Google Scholar 

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

    Google Scholar 

  8. E. F. Mischenko and N. Kh. Rozov, Differential Equations with Small Parameter and Relaxational Oscillations, [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  9. V. F. Butuzov, “Asymptotics of a solution of the equation µ2 Δu − k2(x, y)u = f(x, y) in a rectangular domain,” Differents. Uravn.,9, No. 9, 1654–1660 (1973).

    Google Scholar 

  10. V. F. Butuzov, “A singularly perturbed equation of parabolic type,” Vestn. Mosk. Gos. Univ. Ser. Vychisl. Mat. Kibern., No. 2, 49–56 (1978).

    Google Scholar 

  11. V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  12. S. A. Kashchenko, “Limit values of eigenvalues of the first boundary problem for a singularly perturbed second order equation with rotation points,” Vestn. YarGU, No. 10, 3–39.

  13. A. Stokes, “On the approximation of nonlinear oscillation,” in: Proceedings of the Fifth International Conference on Nonlinear Oscillations [in Russian], Vol. 2, Naukova Dumka, Kiev (1970), pp. 480–491.

    Google Scholar 

  14. V. I. Yudovich, “Methods of linearization in problems of stability of periodic motions of a viscous fluid,” in: Proceedings of the Sixth Winter School on Mathematical Programming and Related Questions, Drogobych, 1973 [in Russian], TsÉMI, Moscow (1975), pp. 44–113.

    Google Scholar 

  15. A. M. Il'in, A. S. Kalashnikov, and O. A. Oleinik, “Second-order linear equations of parabolic type,” Usp. Mat. Nauk,27, No. 3, (105), 3–46 (1962).

    Google Scholar 

  16. B. M. Levitan and V. V. Shikov, Almost Periodic Functions and Differential Equations [in Russian], Moscow State Univ. (1978).

  17. E. P. Sobolevskii, “Estimates of noncommutingness in the averaging principle,” Dokl. Akad. Nauk SSSR,248, No. 3, 552–555 (1984).

    Google Scholar 

Download references

Authors

Additional information

Yaroslavl'. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 27, No. 6, pp. 116–127, November–December, 1986.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kashchenko, S.A. Asymptotics of periodic solutions of autonomous parabolic equations with small diffusion. Sib Math J 27, 880–889 (1986). https://doi.org/10.1007/BF00970006

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00970006

Keywords

Navigation