Skip to main content
SpringerLink
Log in

Degenerate Higher-Order Ordinary Differential Equations and Some of Their Applications

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

Abstract

The article consists of two parts. In the first of them, the solvability of boundary value problems for degenerate higher-order ordinary differential equations is studied. Existence and uniqueness theorems for regular solutions (solutions having all weak derivatives in the sense of S.L. Sobolev occurring in the corresponding equation) are proved. The results obtained are applied in the second part of the article to study the solvability of boundary value problems for some classes of differential equations not solved for the derivative with respect to the distinguished variable.

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

REFERENCES

  1. M. V. Keldysh, ‘‘On some cases of degeneration of equations of elliptic type,’’ Dokl. Akad. Nauk SSSR 77, 181–183 (1951).

    Google Scholar 

  2. G. Fichera, ‘‘On a unified theory of boundary value problems for elliptic-parabolic equations of second order,’’ in Boundary Problems in Differential Equations (Univ. of Wisconsin Press, Madison, 1960), pp. 97–107.

    MATH  Google Scholar 

  3. O. A. Oleinik and E. V. Radkevich, Equations with Non-Negative Characteristic Form (Mosk. Gos. Univ., Moscow, 2010) [in Russian].

    Google Scholar 

  4. V. N. Vragov, ‘‘On the theory of boundary value problems for mixed-type equations,’’ Differ. Uravn. 13, 1098–1105 (1977).

    Google Scholar 

  5. I. E. Egorov and V. E. Fedorov, Nonclassical Highest Order Equations of Mathematical Physics (Vychisl. Tsentr SO RAN, Novosibirsk, Russia, 1995) [in Russian].

    Google Scholar 

  6. I. E. Egorov and V. E. Fedorov, ‘‘Smooth solutions of parabolic equations with changing time direction,’’ AIP Conf. Proc. 2048, 040013 (2018).

    Article  Google Scholar 

  7. A. I. Kozhanov, ‘‘Nonlocal integro-differential equations of the second order with degeneration,’’ Mathematics 8, 606 (2020).

    Article  Google Scholar 

  8. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Am. Math. Soc., Providence, 1991).

    MATH  Google Scholar 

  9. O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and QuasiLinear Elliptic Equations (Academic, New York, 1968).

    MATH  Google Scholar 

  10. H. Triebel, Interpolation Theory. Function Spaces. Differential Operators (VEB Deutscher Verlag der Wissenschaften, Berlin, 1978).

    MATH  Google Scholar 

  11. M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969; Dover, New York, 2014).

  12. A. I. Kozhanov, ‘‘On boundary value problems for some classes of equations of high order not resolved with respect to the highest derivative,’’ Sib. Math. Zh. 35, 359–376 (1994).

    Google Scholar 

  13. A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces (Marcel Dekker, New York, 1999).

    MATH  Google Scholar 

  14. G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems Not Solvable with Respect to Highest Order Derivatives (Marcel Dekker, New York, 2003).

    Book  Google Scholar 

  15. G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators (VSP, Utrecht, The Netherlands, 2003).

    Book  Google Scholar 

  16. A. A. Zamyshlyaeva, High-Order Linear Equations of Sobolev Type (Yuzh.-Ural. Gos. Univ., Chelyabinsk, 2012) [in Russian].

    MATH  Google Scholar 

  17. A. A. Zamyshlyaeva, ‘‘Investigation of high-order linear models of Sobolev type,’’ Doctoral (Phys. Math.) Dissertation (Chelyabinsk, 2013).

  18. V. I. Zhegalov, A. N. Mironov, and E. A. Utkina, Equations with a Dominant Partial Derivative (Kazan Fed. Univ., Kazan, 2014) [in Russian].

    Google Scholar 

  19. A. I. Kozhanov, ‘‘Boundary value problems for a class of nonlocal integro-differential equations with degeneration,’’ Vestn. Samar. Univ., Estestv.-Nauch. Ser. 23 (4), 19–24 (2017).

    Google Scholar 

  20. L. C. Evans, Partial Differential Equations (Am. Math. Soc., Providence, RI, 1998).

    MATH  Google Scholar 

Download references

Funding

The study was carried out within the framework of the state contact of the Sobolev Institute of Mathematics (project no. 0314–2019–0010).

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. I. Kozhanov

Authors
  1. A. I. Kozhanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Kozhanov.

Ethics declarations

The author declares no conflict of interest. The funders had no role in design of the study; in the collection, analysis, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Additional information

(Submitted by T. K. Yuldashev)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kozhanov, A.I. Degenerate Higher-Order Ordinary Differential Equations and Some of Their Applications. Lobachevskii J Math 43, 219–228 (2022). https://doi.org/10.1134/S199508022204014X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S199508022204014X

Keywords:

Search

Navigation