Geodesic
On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity
Sbornik. Mathematics, Tome 53 (1986) no. 2, pp. 399-410 
A. S. Kalashnikov. On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity. Sbornik. Mathematics, Tome 53 (1986) no. 2, pp. 399-410. http://geodesic.mathdoc.fr/item/SM_1986_53_2_a6/
@article{SM_1986_53_2_a6,
     author = {A. S. Kalashnikov},
     title = {On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity},
     journal = {Sbornik. Mathematics},
     pages = {399--410},
     year = {1986},
     volume = {53},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1986_53_2_a6/}
}
TY  - JOUR
AU  - A. S. Kalashnikov
TI  - On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity
JO  - Sbornik. Mathematics
PY  - 1986
SP  - 399
EP  - 410
VL  - 53
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1986_53_2_a6/
LA  - en
ID  - SM_1986_53_2_a6
ER  - 
%0 Journal Article
%A A. S. Kalashnikov
%T On the dependence of properties of solutions of parabolic equations in unbounded domains on the behavior of the coefficients at infinity
%J Sbornik. Mathematics
%D 1986
%P 399-410
%V 53
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1986_53_2_a6/
%G en
%F SM_1986_53_2_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

For linear parabolic equations of second order it is proved that the solution of the first boundary value problem may remain bounded at interior points in spite of the fact that the boundary function tends to infinity together with the time variable if there are lower order terms having definite signs and increasing sufficiently fast in absolute value. For quasilinear parabolic (possibly degenerate) equations of second order it is established that decay of the lower order coefficients as the spatial coordinates tend to infinity may entail the disappearance of effects of total stabilization in finite time and of instantaneous compactification of the support of the solution. Bibliography: 11 titles.

[1] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., “Lokalizatsiya protsessov v srede s postoyannymi svoistvami”, DAN SSSR, 247:2 (1979), 349–353 | MR

[2] Tikhonov A. H., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1977

[3] Evans L. C, Knerr B. F., “Instantaneous shrinking of the support of nonegative solutions to certain nonlinear parabolic equations and variational inequalities III”, J. Math., 23:1 (1979), 153–166 | MR | Zbl

[4] Kalashnikov A. S., “Ob uravnenii teploprovodnosti v srede s neravnomerno raspredelennymi nelineinymi istochnikami ili poglotitelyami tepla”, Vestn. MGU. Matem., mekh., 1983, no. 3, 20–24 | MR | Zbl

[5] Kalashnikov A. S., “O vliyanii pogloscheniya na rasprostranenie tepla v srede s tepploprovodnostyu, zavisyaschei ot temperatury”, ZhVM i MF, 16:3 (1976), 689–696 | MR | Zbl

[6] Kersner R., “Degenerate parabolic equations with general nonlinearities”, Nonlinear Analysis, 4:6 (1980), 1043–1062 | DOI | MR | Zbl

[7] Herrero M. A., “Sobre el comportamiento de las soluciones de ciertos problemas parabolicos no lineales”, Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid, 75:5 (1981), 1165–1183 | MR | Zbl

[8] Pokrovskii JI. D., Taranenko S. N., “Ob usloviyakh prostranstvennoi lokalizatsii reshenii nelineinogo uravneniya teploprovodnosti”, ZhVM i MF, 22:3 (1982), 747–751 | MR | Zbl

[9] Diaz J. I., Veron L., “Compacité du support des solutions d'équations quasi linéaires elliptiques ou paraboliques”, Comptes Rendus Acad. Sci. Ser. 1, 297:3 (1983), 149–152 | MR | Zbl

[10] Granik I. S., “K voprosu o lokalizatsii temperaturnykh vozmuschenii v sredakh s ob'emnym pogloscheniem tepla”, ZhVM i MF, 18:3 (1978), 770–774. | MR | Zbl

[11] Ilin A. M., Kalashnikov A. S, Oleinik O. A., “Lineinye uravneniya vtorogo poryadka parabolicheskogo tipa”, UMN, 17:3 (1962), 3–146 | MR | Zbl