Geodesic
On the boundary values of solutions of parabolic equations
Sbornik. Mathematics, Tome 32 (1977) no. 3, pp. 347-370 
I. M. Petrushko. On the boundary values of solutions of parabolic equations. Sbornik. Mathematics, Tome 32 (1977) no. 3, pp. 347-370. http://geodesic.mathdoc.fr/item/SM_1977_32_3_a5/
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Necessary and sufficient conditions are established for a solution of a second order parabolic equation in a cylinder with smooth lateral boundary to have a limit in the mean on the lateral surface. Bibliography: 2 titles.

[1] V. P. Mikhailov, “O granichnykh znacheniyakh reshenii ellipticheskikh uravnenii v oblastyakh s gladkoi granitsei”, Matem. sb., 101 (143) (1976), 163–188

[2] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, izd-vo «Nauka», Moskva, 1967