Abstract
We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.
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I. G. Petrovskii, “On the solution of the first boundary value problem for the heat equation,”Uch. Zapiski Moskov. Gos. Univ., 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, “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 problem for parabolic and degenerate elliptic equations,”Mat. Sb. [Math. USSR-Sb.],82 (124), No. 4(8), 551–573 (1970).
O. A. Oleinik and E. V. Radkevich,Second-Order Equatons With Nondegenerate Characteristic Form [in Russian], Itogi Nauki i Tekhniki. Matem. Analiz, VINITI, Moscow (1969).
S. D. Ivasishen, “Estimates of the Green function of the homogeneous first boundary value problem for a parabolic equation in a noncylindrical domain,”Ukrain. Mat. Zh. [Ukrainian Math. J.],21, No. 1, 15–27 (1969).
V. A. Solonnikov,On Boundary Value Problems for Linear Parabolic Systems of Differential Equations of the General Form [in Russian], Vol. 83, Trudy Steklov Mathematics Institute, Nauka, Moscow-Leningrad (1965).
V. A. Kozlov and V. G. Maz'ya, “Singularities of the solution of the first boundary value problem for the heat equation in domains with conical points,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], No. 3, 37–44 (1987).
Van Ngok Doan, “Asymptotics of solutions of boundary value problems for second-order parabolic equations in a neighborhood of a corner point of the boundary,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 1, 34–36 (1984).
V. N. Aref'ev and L. A. Bagirov, “Asymptotic behavior of solutions of the Dirichlet problem in domains with singularities,”Mat. Zametki [Math. Notes],59, No. 1, 12–23 (1996).
V. N. Aref'ev and L. A. Bagirov,Asymptotic Behavior of Solutions of the Diffusion Equation in Domains With Singularities [in Russian], Preprint No.563, IPM RAS, Moscow (1996).
H. Bateman and A. Erdélyi,Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York-Toronto-London (1953).
L. A. Bagirov, “A priori estimates, existence theorems, and behavior at infinity of solutions to semielliptic equations in ℝn,”Mat. Sb. [Math. USSR-Sb.],110 (152), No. 4(12), 475–492 (1979).
M. S. Agranovich and M. I. Vishik, “Elliptic problems with parameter and parabolic problems of the general form,”Uspekhi Mat. Nauk [Russian Math. Surveys],19, No. 3, 53–161 (1964).
P. M. Blekher, “On operatos meromorphically depending on a parameter,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 5, 30–36 (1969).
A. Milne,Proc. Edinburgh Math. Soc.,33, 48–64 (1915).
L. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva,Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
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Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.
This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.
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Aref'ev, V.N., Bagirov, L.A. Solutions of the heat equation in domains with singularities. Math Notes 64, 139–153 (1998). https://doi.org/10.1007/BF02310297
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DOI: https://doi.org/10.1007/BF02310297