Skip to main content
SpringerLink
Log in

Pseudo Hyperbolic Equations with Degeneracy: Existence and Uniqueness of Solutions

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

Abstract

The work is devoted to the study of the solvability of initial-boundary value problems for differential equations

$$h(t)u_{tt}-\left(a\frac{\partial}{\partial t}+b\right)\Delta u+cu=f(x,t)$$

(\(\Delta\) is the Laplace operator in space variables) with a non-negative function \(h(t)\). Similar equations are called pseudohyperbolic equations in the literature. The aim of the work is to prove the existence and uniqueness of regular solutions of the problems under study—solutions that have all S.L. Sobolev derivatives included in the equation.

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

REFERENCES

  1. N. A. Larkin, ‘‘Existence theorems for quasilinear pseudohyperbolic equations,’’ Sov. Math. Dokl. 26, 260–263 (1982).

    Google Scholar 

  2. Y. Liu and H. Li, ‘‘H1-Galerkin mixed finite element methods for pseudo-hyperbolic equations,’’ Appl. Math. Comput. 212, 446–457 (2009).

    MathSciNet  MATH  Google Scholar 

  3. K. I. Khudaverdiyev and A. A. Veliyev, Investigation of a One-Dimensional Mixed Problem for a Class of Third-Order Pseudohyperbolic Equations with a Nonlinear Operator Right-Hand Side (Chashyogly, Baku, 2010) [in Russian].

    Google Scholar 

  4. G. V. Demidenko and S. V. Uspenskii, Equations and Systems Not Resolved with Respect to the Highest Derivative (Nauka, Novosibirsk, 1998) [in Russian].

    Google Scholar 

  5. M. O. Korpusov, Blow-up in the Nonlinear Wave Equations with Positive Energy (URSS, Moscow, 2012) [in Russian].

    MATH  Google Scholar 

  6. A. I. Kozhanov, Composite Type Equations and Inverse Problems (VSP, Utrecht, 1999).

    Book  MATH  Google Scholar 

  7. I. E. Egorov, ‘‘Vragov’s boundary value problem for an implicit equation of mixed type,’’ J. Phys.: Conf. Ser. 894, 012028 (2017).

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

    Google Scholar 

  9. V. N. Vragov, ‘‘On statements and solvability of boundary value problems for equations of mixed composite type,’’ in Mathematical Analysis and Related Questions of Mathematics, Collection of Articles (Nauka, Novosibirsk, 1978), pp. 5–13 [in Russian].

    Google Scholar 

  10. I. E. Egorov and V. E. Fedorov, Nonclassical Equations of High Order Mathematical Physics (Izd. Vychisl. Tsetra SO RAN, Novosibirsk, 1995) [in Russian].

    Google Scholar 

  11. A. I. Kozhanov, ‘‘Initial-boundary value problems for degenerate hyperbolic equations,’’ Sib. Electron. Math. Rep. 18, 43–53 (2021).

    MathSciNet  MATH  Google Scholar 

  12. S. N. Glazatov, ‘‘On the well-posedness of a mixed problem for a degenerate hyperbolic equation with arbitrary character of degeneracy,’’ Sib. Math. J. 28, 224–229 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  13. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Vol. 90 of Translations of Mathematical Monographs (AMS, Providence, RI, 1991).

  14. O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Vol. 46 of Mathematics in Science and Engineering (Academic, New York, 1968).

  15. H. Triebel, Interpolation Theory. Functional Spaces. Differetial Operators (VEB Duetschen Verlag der Wissenschaften, Berlin, 1978).

    Google Scholar 

  16. A. I. Kozhanov, ‘‘A mixed problem for some classes of nonlinear third-order equations,’’ Sb. Math. 46, 507–525 (1983).

    Article  MATH  Google Scholar 

  17. S. Ya. Yakubov, Linear Operator-Differential Equations and their Applications (Ehlm, Baku, 1985) [in Russian].

    MATH  Google Scholar 

  18. V. A. Trenogin, Functional Analysis (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ammosov North-Eastern Federal University, Mirny Polytechnic Institute, 678175, Mirny, Russia

    G. A. Varlamova

  2. Sobolev Institute of Mathematics, 630090, Novosibirsk, Russia

    A. I. Kozhanov

  3. Academy of Science of the Republic of Sakha (Yakutia), 677007, Yakutsk, Russia

    A. I. Kozhanov

Authors
  1. G. A. Varlamova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. I. Kozhanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to G. A. Varlamova or A. I. Kozhanov.

Additional information

(Submitted by A. B. Muravnik)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Varlamova, G.A., Kozhanov, A.I. Pseudo Hyperbolic Equations with Degeneracy: Existence and Uniqueness of Solutions. Lobachevskii J Math 44, 3594–3603 (2023). https://doi.org/10.1134/S1995080223080553

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080223080553

Keywords:

Search

Navigation