Skip to main content
SpringerLink
Log in

On asymptotics of solutions of parabolic equations with nonlocal high-order terms

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In the paper, we study the Cauchy problem for second-order differential-difference parabolic equations containing translation operators acting to the high-order derivatives with respect to spatial variables. We construct the integral representation of the solution and investigate its long-term behavior. We prove theorems on asymptotic closeness of the constructed solution and the Cauchy problem solutions for classical parabolic equations; in particular, conditions of the stabilization of the solution are obtained.

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. R. Bellman and K. Cooke, Differential-Difference Equations, Academic Press, New York (1963).

    Google Scholar 

  2. J. Hale, Theory of Functional Differential Equations, Springer, New York (1984).

    Google Scholar 

  3. A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhauser, Basel (1997).

    Google Scholar 

  4. K. Kunisch and W. Schappacher, “Necessary conditions for partial differential equations with delay to generate C 0-semigroups,” J. Differential Equations, 50, No. 1, 49–79 (1983).

    Article  MathSciNet  Google Scholar 

  5. W. Desch and W. Schappacher, “Spectral properties of finite-dimensional perturbed linear semigroups,” J. Differential Equations, 59, No. 1, 80–102 (1985).

    Article  MathSciNet  Google Scholar 

  6. V. V. Vlasov, “On a class of differential-difference equations in a Hilbert space and some spectral questions,” Russian Acad. Sci. Dokl. Math., 46, No. 3, 458–462 (1993).

    MathSciNet  Google Scholar 

  7. V. M. Borok and E. S. Viglin, “The uniqueness of the solution of the fundamental initial problem for partial differential equations with a deviating argument,” Differential Equations, 13, No. 7, 848–854 (1977).

    MathSciNet  Google Scholar 

  8. A. I. Daševskii, “A boundedness criterion for the solutions of linear difference-differential equations with retarded argument in Banach spaces,” Differential Equations, 13, No. 8, 1054–1056 (1977).

    Google Scholar 

  9. A. Inone, T. Miyakawa, and K. Yoshida, “Some properties of solutions for semilinear heat equations with time lag,” J. Differential Equations, 24, No. 3, 383–396 (1977).

    MathSciNet  Google Scholar 

  10. G. Di Blasio, K. Kunisch, and E. Sinestrari, “L 2-regularity for parabolic partial integrodifferential equations with delay in highest-order derivatives,” J. Math. Anal. Appl., 102, No. 1, 38–57 (1984).

    MathSciNet  Google Scholar 

  11. V. V. Vlasov and V. Zh. Sakbaev, “The correct solvability of some differential-difference equations in the scale of Sobolev spaces,” Differential Equations, 37, No. 9, 1252–1260 (2001).

    Article  MathSciNet  Google Scholar 

  12. B. L. Gurevič, “New types of fundamental and generalized function spaces and the Cauchy problem for systems of difference equations involving differential operations,” Dokl. Akad. Nauk SSSR, 108, No. 6, 1001–1003 (1956).

    MathSciNet  Google Scholar 

  13. V. S. Rabinovich, “The Cauchy problem for parabolic differential-difference operators with variable coefficients,” Differential Equations, 19, No. 6, 768–775 (1983).

    MATH  MathSciNet  Google Scholar 

  14. A. L. Skubachevskii, “On some properties of elliptic and parabolic differential-difference equations,” Russ. Math. Surv., 51, No. 1, 169–170 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  15. A. L. Skubachevskii, “Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics,” Nonlinear Anal., 32, No. 2, 261–278 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  16. A. L. Skubachevskii and R. V. Shamin, “The first mixed problem for a parabolic differential-difference equation,” Math. Notes, 66, No. 1–2, 113–119 (1999).

    MathSciNet  Google Scholar 

  17. A. B. Muravnik, “On Cauchy problem for parabolic differential-difference equations,” Nonlinear Anal., 51, No. 2, 215–238 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  18. A. B. Muravnik, “On the Cauchy problem for differential-difference equations of the parabolic type,” Russian Acad. Sci. Dokl. Math., 66, No. 1, 107–110 (2002).

    Google Scholar 

  19. A. V. Razgulin, “Rotational multi-petal waves in optical system with 2-D feedback,” Chaos in Optics. Proceedings SPIE, 2039, 342–352 (1993).

    Google Scholar 

  20. M. A. Vorontsov, N. G. Iroshnikov, and R. L. Abernathy, “Diffractive patterns in a nonlinear optical two-dimensional feedback system with field rotation,” Chaos, Solitons, and Fractals, 4, 1701–1716 (1994).

    Article  Google Scholar 

  21. I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. 3. Theory of Differential Equations, Academic Press, New York (1967).

    Google Scholar 

  22. V. M. Borok and Ja. I. Zitomirskii, “On the Cauchy problem for linear partial differential equations with linearly transformed argument,” Sov. Math. Dokl., 12, 1412–1416 (1971).

    Google Scholar 

  23. O. A. Ladyzhenskaya, “On the uniqueness of the Cauchy problem solution for a linear parabolic equation,” Mat. Sb., 27 (59), No. 2, 175–184 (1950).

    Google Scholar 

  24. V. D. Repnikov and S. D. Ehjdel’man, “Necessary and sufficient conditions for the establishment of a solution of the Cauchy problem,” Sov. Math. Dokl., 7, 388–391 (1966).

    Google Scholar 

  25. M. E. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton (1981).

    Google Scholar 

  26. A. M. Il’in, A. S. Kalašnikov, and O. A. Oleinik, “Second-order linear equations of parabolic type,” Russian Math. Surv., 17, No. 3, 1–146 (1962).

    Google Scholar 

Download references

Authors
  1. A. B. Muravnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 143–183, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Muravnik, A.B. On asymptotics of solutions of parabolic equations with nonlocal high-order terms. J Math Sci 135, 2695–2720 (2006). https://doi.org/10.1007/s10958-006-0139-1

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-006-0139-1

Keywords

Search

Navigation