Geodesic
On the behavior, for large time values, of nonnegative solutions of the second mixed problem for a parabolic equation
Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 195-209 
A. V. Lezhnev. On the behavior, for large time values, of nonnegative solutions of the second mixed problem for a parabolic equation. Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 195-209. http://geodesic.mathdoc.fr/item/SM_1987_57_1_a12/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The author studies the behavior, for large time values $t$, of a nonnegative solution of the second mixed problem for a uniformly parabolic equation $$ \frac{\partial u(x,t)}{\partial t}=\sum_{i,j=1}^n\frac\partial{\partial x_i}\biggl(a_{ij}(x,t)\frac{\partial u(x,t)}{\partial x_j}\biggr) $$ in a cylindrical domain $\Omega\times\{t>0\}$, where $\Omega$ is an unbounded domain in $\mathbf R^n$. It is shown that for a certain class of unbounded domains $\Omega$, the behavior of the solution of the problem as $t\to\infty$ is determined by the behavior, for large values of the parameter $R$, of the means of the initial function over the sets $\{x\in\Omega:|x-\xi|, $\xi\in\Omega$, $R>0$. Bibliography: 8 titles.

[1] Zelenyak T. I., Mikhailov V. P., “Asimptoticheskoe povedenie reshenii nekotorykh kraevykh zadach matematicheskoi fiziki pri $t\to\infty$”, Differentsialnye uravneniya s chastnymi proizvodnymi, Nauka, M., 1970, 96–118

[2] Guschin A. K., “O ravnomernoi stabilizatsii reshenii vtoroi smeshannoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 119(161) (1982), 451–508

[3] Guschin A. K., “Ob otsenkakh reshenii kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka”, Tr. MIAN, CXXVI (1973), 5–45

[4] Guščin A. K., “On the behaviour as $t\to\infty$ of solutions of the second mixed problem for a second-order parabolic equation”, Appl. Math. Optim., 6 (1980), 169–180 | DOI | MR

[5] Guschin A. K., “Stabilizatsiya reshenii vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, Matem. sb., 101(143) (1976), 459–499 | Zbl

[6] Guschin A. K., Mikhailov V. P., Mikhailov Yu. A., “Otsenki funktsii Grina i kriterii ravnomernoi stabilizatsii resheniya vtoroi smeshannoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, DAN SSSR, 282:1 (1985), 15–18 | MR | Zbl

[7] Oleinik O. A., “O edinstvennosti reshenii kraevykh zadach i zadachi Koshi dlya obschikh parabolicheskikh sistem”, DAN SSSR, 220:6 (1975), 34–37

[8] Guschin A. K., “Nekotorye svoistva obobschennogo resheniya vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 97(139) (1975), 242–261 | Zbl