Skip to main content
SpringerLink
Log in

A study of the stability of solutions of linear parabolic equations with nearly constant coefficients and small diffusion

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

We study the stability of the solutions of boundary value problems for a certain class of Petrovskii-parabolic systems with sufficiently small diffusion coefficients. The dimension of the set of critical cases in the stability problem turns out to be infinite. We develop an efficient algorithm for studying stability. As examples we consider parabolic boundary value problems with delay and rapidly oscillating coefficients, the problem of parametric resonance under a double-frequency perturbation, and problems with variable leading terms and variable domain of definition. Bibliography: 21 titles.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. Yu. S. Kolesov, “On the stability of solutions of linear differential equations of parabolic type with almost-periodic coefficients,”Tr. Mosk. Mat. Obshch.,36, 3–27 (1978).

    Google Scholar 

  2. O. A. Oleinik, “On averaging differential equations with rapidly oscillating coefficients,” in:Analytic Methods of the Theory of Nonlinear Oscillations [in Russian], Vol. 1, Naukova Dumka, Kiev (1984), pp. 286–292.

    Google Scholar 

  3. V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, “G-convergence of parabolic operators,”Usp. Mat. Nauk,36, No. 1, 11–58 (1981).

    Google Scholar 

  4. B. M. Levitan and V. V. Zhikov,Almost-periodic Functions and Differential Equations, New York, Cambridge University Press (1982).

    Google Scholar 

  5. A. B. Vasil'eva, S. V. Dvoryaninov, and N. Kh. Rozov, “Toward an asymptotic theory of oscillations of singularly perturbed parabolic systems,” in:Ninth Int. Conf. Nonlinear Osc. [in Russian], Vol. 1, Naukova Dumka, Kiev (1984), pp. 107–111.

    Google Scholar 

  6. A. B. Vasil'eva, S. A. Kashchenko, Yu. S. Kolesov, and N. Kh. Rozov, “Bifurcation of the self-oscillations of nonlinear parabolic equations with small diffusion,”Mat. Sb.,130, No. 4, 488–499 (1986).

    Google Scholar 

  7. I. Z. Shtokalo,Operational Calculus [in Russian], Naukova Dumka, Kiev (1972).

    Google Scholar 

  8. Yu. S. Kolesov, V. S. Kolesov, and I. I. Fedik,Self-oscillations in Systems with Distributed Parameters [in Russian], Naukova Dumka, Kiev (1979).

    Google Scholar 

  9. Yu. S. Kolesov and V. V. Maiorov, “A new method of studying the stability of solutions of linear differential equations with nearly constant almost-periodic coefficients,”Differents. Uravn.,10, No. 10, 1778–1788 (1974).

    Google Scholar 

  10. V. I. Arnol'd, “Lectures on Bifurcations and Versal Families,”Usp. Mat. Nauk,27, No. 5, 119–184 (1972).

    Google Scholar 

  11. Yu. S. Kolesov and B. P. Kubyshkin, “A minimal algorithm for studying the stability of linear systems,” in:Studies in Stability and Theory of Oscillations [in Russian], Yaroslavl' (1977), pp. 142–155.

  12. Yu. S. Kolesov, “Periodic solutions of quasilinear second-order parabolic equations,”Tr. Mosk. Mat. Obshch.,21, 103–134 (1970).

    Google Scholar 

  13. V. F. Chaplygin, “Positive periodic solutions of singularly perturbed nonlinear second-order differential equations,”Tr. Nauch.-Issl. Inst. Mat. VGU, No. 2, 43–46 (1970).

    Google Scholar 

  14. S. A. Kashchenko, “Limiting values of eigenvalues of the initial/boundary value problem for a singularly perturbed second-order differential equation with turning points,”Vest. Yarosl. Univ., No. 10, 3–39 (1974).

    Google Scholar 

  15. S. A. Kashchenko, “The asymptotics of the eigenvalues of periodic and antiperiodic boundary value problems for singularly perturbed second-order differential equations with turning points,”Vest. Yarosl. Univ., No. 13, 20–84 (1975).

    Google Scholar 

  16. A. M. Il'in, A. S. Kalashnikov, and O. A. Oleinik, “Linear second-order equations of parabolic type,”Usp. Mat. Nauk,XVII, No. 3, 3–146 (1962).

    Google Scholar 

  17. S. A. Kashchenko and V. V. Maiorov, “An algorithm for studying the stability of the solutions of linear differential equations with lag and rapidly oscillating almost-periodic coefficients,” in:Studies in Stability and the Theory of Oscillations [in Russian], Yaroslavl' (1977), pp. 70–81.

  18. S. A. Kashchenko and Yu. S. Kolesov, “Parametric resonance in systems with delay under a double-frequency perturbation,”Sib. Mat. Zh.,XXI, No. 2, 113–118 (1980).

    Google Scholar 

  19. S. A. Kashchenko and Yu. S. Kolesov, “Damping of ‘swings’ using a double-frequency force,” in:Studies in Stability and the Theory of Oscillations [in Russian], Yaroslavl' (1977), pp. 19–25.

  20. V. N. Fomin,The Mathematical Theory of Parametric Resonance in Linear Distributed Systems [in Russian], Leningrad University Press (1972).

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

    Google Scholar 

Download references

Authors
  1. S. A. Kashchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 128–155, 1991.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kashchenko, S.A. A study of the stability of solutions of linear parabolic equations with nearly constant coefficients and small diffusion. J Math Sci 60, 1742–1764 (1992). https://doi.org/10.1007/BF01102587

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01102587

Keywords

Search

Navigation