Geodesic
Limit passage in quasilinear parabolic equations with weakly converging coefficients, and the asymptotic behavior of solutions of the Cauchy problem
Sbornik. Mathematics, Tome 70 (1991) no. 2, pp. 467-484 
V. L. Kamynin. Limit passage in quasilinear parabolic equations with weakly converging coefficients, and the asymptotic behavior of solutions of the Cauchy problem. Sbornik. Mathematics, Tome 70 (1991) no. 2, pp. 467-484. http://geodesic.mathdoc.fr/item/SM_1991_70_2_a8/
@article{SM_1991_70_2_a8,
     author = {V. L. Kamynin},
     title = {Limit passage in quasilinear parabolic equations with weakly converging coefficients, and the asymptotic behavior of solutions of the {Cauchy} problem},
     journal = {Sbornik. Mathematics},
     pages = {467--484},
     year = {1991},
     volume = {70},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1991_70_2_a8/}
}
TY  - JOUR
AU  - V. L. Kamynin
TI  - Limit passage in quasilinear parabolic equations with weakly converging coefficients, and the asymptotic behavior of solutions of the Cauchy problem
JO  - Sbornik. Mathematics
PY  - 1991
SP  - 467
EP  - 484
VL  - 70
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1991_70_2_a8/
LA  - en
ID  - SM_1991_70_2_a8
ER  - 
%0 Journal Article
%A V. L. Kamynin
%T Limit passage in quasilinear parabolic equations with weakly converging coefficients, and the asymptotic behavior of solutions of the Cauchy problem
%J Sbornik. Mathematics
%D 1991
%P 467-484
%V 70
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1991_70_2_a8/
%G en
%F SM_1991_70_2_a8

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

Asymptotic closeness as $t\to+\infty$ (for each $x\in R^n$) is proved for solutions of two distinct Cauchy problems for quasilinear parabolic equations under the condition that certain limit means of the difference of the coefficients and of the difference of the initial functions are equal to zero. This proof is based on reducing the initial problem to a problem on the passage to the limit in a sequence of equations with weakly converging coefficients which is also of independent interest.

[1] Eidelman S. D., Parabolicheskie sistemy, Nauka, M., 1964 | MR

[2] Guschin A. K., Mikhailov V. P., “O stabilizatsii resheniya zadachi Koshi dlya parabolicheskogo uravneniya”, Differents. uravneniya, 7:2 (1971), 297–311 | Zbl

[3] Guschin A. K., Mikhailov V. P., O stabilizatsii resheniya zadachi Koshi dlya parabolicheskogo uravneniya s odnoi prostranstvennoi peremennoi, Tr. MIAN, 112, 1971 | Zbl

[4] Zhikov V. V., “O stabilizatsii reshenii parabolicheskikh uravnenii”, Matem. sb., 104(146) (1977), 597–616 | Zbl

[5] Zhikov V. V., “Kriterii potochechnoi stabilizatsii dlya uravnenii vtorogo poryadka s pochti periodicheskimi koeffitsientami”, Matem. sb., 110(152):2(10) (1979), 304–318 | MR | Zbl

[6] Zhikov V. V., “Asimptoticheskoe povedenie i stabilizatsiya reshenii parabolicheskogo uravneniya s mladshimi chlenami”, Tr. MMO, 46, 1983, 69–98 | MR | Zbl

[7] Kamin S., “On stabilization of solution of the Cauchy problem for parabolic equation”, Proc. Roy. Soc. Edinburg, 76:1 (1976), 43–53 | MR | Zbl

[8] Sorokina A. G., “O blizosti reshenii zadachi Koshi dlya parabolicheskikh uravnenii vtorogo poryadka”, Matem. zametki, 34:1 (1983), 113–121 | MR | Zbl

[9] Porper F. O., Eidelman S. D., “Asimptoticheskoe povedenie klassicheskikh i obobschennykh reshenii odnomernykh parabolicheskikh uravnenii vtorogo poryadka”, Tr. MMO, 36 (1978), 85–130 | MR | Zbl

[10] Porper F. O., Eidelman S. D., “Teoremy ob asimptoticheskoi blizosti reshenii mnogomernykh parabolicheskikh uravnenii vtorogo poryadka”, UMN, 35:1 (1980), 211–212 | MR | Zbl

[11] Repnikov V. D., “O stabilizatsii reshenii parabolicheskikh uravnenii s ostsilliruyuschimi koeffitsientami”, Differents. uravneniya, 23:8 (1987), 1353–1359 | MR | Zbl

[12] Repnikov V. D., “Ob asimptoticheskoi blizosti i stabilizatsii resheniya zadachi Koshi”, Differents. uravneniya, 24:1 (1988), 146–155 | MR | Zbl

[13] Denisov V. N., Zhikov V. V., “O stabilizatsii resheniya zadachi Koshi dlya parabolicheskikh uravnenii”, Matem. zametki, 37:6 (1985), 834–850 | MR | Zbl

[14] Valikov K. V., “Blizost reshenii zadachi Koshi dlya nekotorykh parabolicheskikh uravnenii vtorogo poryadka”, Differents. uravneniya, 23:4 (1987), 686–696 | MR

[15] Nash J., “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80:4 (1958), 931–954 | DOI | MR | Zbl

[16] Markov V. G., Oleinik O. A., “O rasprostranenii tepla v odnomernykh dispersnykh sredakh”, Prikl. matem. i mekhan., 39:6 (1975), 1073–1081 | MR | Zbl

[17] Zhikov V. V., Kozlov S. M., Oleinik O. A., “O $G$-skhodimosti parabolicheskikh operatorov”, UMN, 36:1 (1981), 11–56 | MR

[18] Kruzhkov S. N., Kamynin V. L., “O predelnom perekhode v kvazilineinykh parabolicheskikh uravneniyakh”, Tr. MIAN, 167 (1985), 183–206 | MR | Zbl

[19] Kamynin V. L., “O predelnom perekhode v kvazilineinykh ellipticheskikh uravneniyakh so mnogimi nezavisimymi peremennymi”, Matem. sb., 132(174) (1987), 45–63 | MR

[20] Kamynin V. L., “O nepreryvnoi zavisimosti otnositelno izmenenii koeffitsientov v slabykh normakh reshenii parabolicheskikh uravnenii”, Analiz matematicheskikh modelei fizicheskikh protsessov, Sb. nauchnykh trudov, Energoatomizdat, M., 1988

[21] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[22] Oleinik O. A., Kruzhkov S. N., “Kvazilineinye parabolicheskie uravneniya vtorogo poryadka so mnogimi nezavisimymi peremennymi”, UMN, 16:5 (1961), 115–155 | MR | Zbl

[23] Smirnov V. I., Kurs vysshei matematiki, t. 5, Fizmatgiz, M., 1959 | MR

[24] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR