Skip to main content
SpringerLink
Log in

Uniqueness of the Solution of the Cauchy Problem for Parabolic Systems

  • Partial Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We consider the Cauchy problem for a second-order Petrovskii parabolic system with bounded continuous coefficients under the condition that the leading coefficients are Dini continuous in the spatial variables. We prove the uniqueness of the classical solution of this problem in the space of functions increasing with respect to the spatial variables, belonging to the Tikhonov class, and having derivatives that may be unbounded when approaching the initial data plane.

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. Holmgren, E., Sur les solutions quasianalytiques de l’éequation de la chaleur, Arkiv. Math., 1924, vol. 18, pp. 64–95.

    Google Scholar 

  2. Tychonoff, A., Théeorèmes d’unicitée pour l’éequation de la chaleur, Mat. Sb., 1935, vol. 42, no. 2, pp. 199–216.

    MATH  Google Scholar 

  3. T¨acklind S., Sur les classes quasianalytiques des solutions des éequations aux deriveées partielles du type parabolique, Nord. Acta Regial. Soc. Schientiarum. Uppsaliensis, 1936, vol. 10, ser. 4, no. 3, pp. 3–55.

    Google Scholar 

  4. Il’in, A.M., Kalashnikov, A.S., and Oleinik, O.A., Linear equations of the second order of parabolic type, Russ. Math. Surv., 1962, vol. 17, no. 3 (105), pp. 1–143.

    Article  MATH  Google Scholar 

  5. Oleinik, O.A. and Radkevich, E.V., Method of introducing a parameter for studying evolution equations, Usp. Mat. Nauk, 1978, vol. 33, no. 5, pp. 7–76.

    MATH  Google Scholar 

  6. Kamynin, L.I. and Khimchenko, B.N., On the Tikhonov–Petrovskii problem for second-order parabolic equations, Sib. Mat. Zh., 1981, vol. 22, no. 5, pp. 78–109.

    MathSciNet  MATH  Google Scholar 

  7. Krzyzanski, M., Sur les solutions des éequations du type parabolique déeterminéees dans une réegion illimitéee, Bull. Amer. Math. Soc., 1941, vol. 47, pp. 911–915.

    Article  MathSciNet  Google Scholar 

  8. Petrovskii, I.G., On some problems in the theory of partial differential equations, Usp. Mat. Nauk, 1946, vol. 1, nos. 3–4, pp. 44–70.

    Google Scholar 

  9. Ladyzhenskaya, O.A., On the uniqueness of the solution of the Cauchy problem for a linear parabolic equation, Mat. Sb., 1950, vol. 27 (69), no. 2, pp. 175–184.

    MathSciNet  Google Scholar 

  10. Gel’fand, I.M. and Shilov, G.E., Fourier transform of rapidly growing functions and uniqueness for the Cauchy problem, Usp. Mat. Nauk, 1953, vol. 8, no. 6, pp. 3–54.

    Google Scholar 

  11. Eidel’man, S.D., On the Cauchy problem for parabolic systems, Dokl. Akad. Nauk SSSR, 1954, vol. 98, no. 6, pp. 913–915.

    MathSciNet  MATH  Google Scholar 

  12. Zolotarev, G.N., On the uniqueness of the solution of the Cauchy problem for Petrovskii parabolic systems, Izv. Vyssh. Uchebn. Zaved. Math., 1958, no. 2, pp. 118–135.

    Google Scholar 

  13. Oleinik, O.A., On the uniqueness of solutions of the Cauchy problem for general parabolic systems in classes of rapidly growing functions, Usp. Mat. Nauk, 1974, vol. 29, no. 5, pp. 229–230.

    Google Scholar 

  14. Eidel’man, S.D., Parabolicheskie sistemy (Parabolic Systems), Moscow: Nauka, 1964.

    Google Scholar 

  15. Baderko, E.A. and Cherepova, M.F., Uniqueness theorem for parabolic Cauchy problem, Appl. Anal., 2016, vol. 95, no. 7, pp. 1570–1580.

    Article  MathSciNet  MATH  Google Scholar 

  16. Baderko, E.A. and Cherepova, M.F., Uniqueness of a solution to the Cauchy problem for parabolic systems, Dokl. Math., 2016, vol. 93, no. 3, pp. 316–317.

    Article  MathSciNet  MATH  Google Scholar 

  17. Petrovsky, I.G., On the Cauchy problem for systems of linear partial differential equations in the domain of nonanalytic functions, Byull. Mosk. Gos. Univ. Sect. A, 1938, vol. 1, no. 7, pp. 1–72.

    Google Scholar 

  18. Dzyadyk, V.K., Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami (Introduction to the Theory of Uniform Approximation of Functions by Polynomials), Moscow: Nauka, 1977.

    MATH  Google Scholar 

  19. Friedman, A., Partial Differential Equations of Parabolic Type, Englewood-Cliffs, 1964. Translated under the title Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Moscow: Mir, 1968.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    E. A. Baderko

  2. National Research University “Moscow Power Engineering Institute,”, Moscow, 111250, Russia

    M. F. Cherepova

Authors
  1. E. A. Baderko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. F. Cherepova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to E. A. Baderko or M. F. Cherepova.

Additional information

Russian Text © The Author(s), 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 6, pp. 822–830.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baderko, E.A., Cherepova, M.F. Uniqueness of the Solution of the Cauchy Problem for Parabolic Systems. Diff Equat 55, 806–814 (2019). https://doi.org/10.1134/S0012266119060077

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266119060077

Search

Navigation