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.
Similar content being viewed by others
References
I. G. Petrovskii, “Solution of a boundary value problem for the heat equation,”Uchenye Zapiski MGU, No. 2, 55–59 (1934).
J. J. Kohn and L. Nirenberg, “Degenerate elliptic-parabolic equations of second order,”Comm. Pure Appl. Math.,20, No. 4, 797–872 (1967).
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).
V. P. Mikhailov, “On the Dirichlet problem for a parabolic equation,”Mat. Sb. [Math. USSR-Sb.],61(103), No. 1, 40–64 (1963).
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).
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).
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).
E. A. Baderko, “A parabolic equation in a simple domain,”Differentsial'nye Uravneniya [Differential Equations],27, No. 1, 17–21 (1991).
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).
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).
O. A. Ladyzhenskaya,Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
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).
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).
Arkerud, “OnL p -estimates for quasi-elliptic boundary problems,”Math. Scand.,24, 141–144 (1969).
P. M. Blekher, “Meromorphic operator families,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 5, 30–36 (1969).
G. Bateman and A. Erdélyi,Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York-Toronto-London (1953).
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 12–23, January, 1996.
This work was partially supported by the Russian Foundation for Basic Research under grant No. 242 93-01-16035.
Rights and permissions
About this article
Cite this article
Aref'ev, V.N., Bagirov, L.A. Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities. Math Notes 59, 10–17 (1996). https://doi.org/10.1007/BF02312460
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02312460