Advertisement

Mathematical Notes

, Volume 59, Issue 1, pp 10–17 | Cite as

Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities

  • V. N. Aref'ev
  • L. A. Bagirov
Article

Abstract

We study solutions of the Dirichlet problem for a second-order parabolic equation with variable coefficients in domains with nonsmooth lateral surface. The asymptotic expansion of the solution in powers of the parabolic distance is obtained in a neighborhood of a singular point of the boundary. The exponents in this expansion are poles of the resolvent of an operator pencil associated with the model problem obtained by “freezing” the coefficients at the singular point. The main point of the paper is in proving that the resolvent is meromorphic and in estimating it. In the one-dimensional case, the poles of the resolvent satisfy a transcendental equation and can be expressed via parabolic cylinder functions.

Keywords

Asymptotic Behavior Singular Point Asymptotic Expansion Parabolic Equation Dirichlet Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. G. Petrovskii, “Solution of a boundary value problem for the heat equation,”Uchenye Zapiski MGU, No. 2, 55–59 (1934).Google Scholar
  2. 2.
    J. J. Kohn and L. Nirenberg, “Degenerate elliptic-parabolic equations of second order,”Comm. Pure Appl. Math.,20, No. 4, 797–872 (1967).MathSciNetGoogle Scholar
  3. 3.
    V. A. Kondrat'ev, “Boundary value problems for parabolic equations in closed domains,”Trudy Moskov. Mat. Obshch. [Trans. Moscow Math. Soc.],15, 400–451 (1967).Google Scholar
  4. 4.
    V. P. Mikhailov, “On the Dirichlet problem for a parabolic equation,”Mat. Sb. [Math. USSR-Sb.],61(103), No. 1, 40–64 (1963).MathSciNetGoogle Scholar
  5. 5.
    V. I. Feigin, “Smoothness of solutions to boundary value problems for parabolic and degenerate elliptic equations,”Mat. Sb. [Math. USSR-Sb.],82, No. 4, 551–573 (1970).zbMATHMathSciNetGoogle Scholar
  6. 6.
    S. D. Ivasishen, “Estimates of the Green functions for the homogeneous Dirichlet problem for a parabolic equation in a nontube domain,”Ukrain. Mat. Zh.,21, No. 1, 15–27 (1969).zbMATHGoogle Scholar
  7. 7.
    V. A. Solonnikov, “Boundary value problems for general linear parabolic systems of differential equations,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],83, 3–162 (1965).MathSciNetGoogle Scholar
  8. 8.
    E. A. Baderko, “A parabolic equation in a simple domain,”Differentsial'nye Uravneniya [Differential Equations],27, No. 1, 17–21 (1991).zbMATHMathSciNetGoogle Scholar
  9. 9.
    M. O. Orynbasarov, “Solvability of boundary value problems for a parabolic and a polyparabolic equation in a nontube domain with nonsmooth lateral surface,”Differentsial'nye Uravneniya [Differential Equations],30, No. 1, 151–161 (1994).zbMATHMathSciNetGoogle Scholar
  10. 10.
    Doan Van Ngok, “Asymptotics of solutions to boundary value problems for second-order parabolic equations in a neighborhood of a corner point on the boundary,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 1, 34–36 (1984).zbMATHGoogle Scholar
  11. 11.
    O. A. Ladyzhenskaya,Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).Google Scholar
  12. 12.
    L. A. Bagirov, “A priori estimates, existence theorems, and behavior at infinity of solutions of quasielliptic equations in ℝn,”Mat. Sb. [Math. USSR-Sb.],110, No. 4, 475–492 (1979).zbMATHMathSciNetGoogle Scholar
  13. 13.
    M. S. Agranovich and M. I. Vishik, “Elliptic problems with a parameter and general parabolic problems,”Uspekhi Mat. Nauk [Russian Math. Surveys],19, No. 3, 53–161 (1964).Google Scholar
  14. 14.
    Arkerud, “OnL p-estimates for quasi-elliptic boundary problems,”Math. Scand.,24, 141–144 (1969).MathSciNetGoogle Scholar
  15. 15.
    P. M. Blekher, “Meromorphic operator families,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 5, 30–36 (1969).Google Scholar
  16. 16.
    G. Bateman and A. Erdélyi,Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York-Toronto-London (1953).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • V. N. Aref'ev
    • 1
  • L. A. Bagirov
    • 2
  1. 1.Moscow State Civil Engineering UniversityUSSR
  2. 2.Institute for Problems in MechanicsRussian Academy of SciencesUSSR

Personalised recommendations