Geodesic
Asymptotics of a fundamental solution of a parabolic equation as  $t\to\infty$
Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 315-337 
E. F. Lelikova. Asymptotics of a fundamental solution of a parabolic equation as  $t\to\infty$. Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 315-337. http://geodesic.mathdoc.fr/item/SM_1988_60_2_a3/
@article{SM_1988_60_2_a3,
     author = {E. F. Lelikova},
     title = {Asymptotics of~a~fundamental solution of~a~parabolic equation as~ $t\to\infty$},
     journal = {Sbornik. Mathematics},
     pages = {315--337},
     year = {1988},
     volume = {60},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1988_60_2_a3/}
}
TY  - JOUR
AU  - E. F. Lelikova
TI  - Asymptotics of a fundamental solution of a parabolic equation as  $t\to\infty$
JO  - Sbornik. Mathematics
PY  - 1988
SP  - 315
EP  - 337
VL  - 60
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1988_60_2_a3/
LA  - en
ID  - SM_1988_60_2_a3
ER  - 
%0 Journal Article
%A E. F. Lelikova
%T Asymptotics of a fundamental solution of a parabolic equation as  $t\to\infty$
%J Sbornik. Mathematics
%D 1988
%P 315-337
%V 60
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1988_60_2_a3/
%G en
%F SM_1988_60_2_a3

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

The author analyzes the behavior as $t\to\infty$ of the fundamental solution $G(x, s, t)$ of the Cauchy problem for the equation $v_t-v_{xx}-a(x)v_x-b(x)v=0$ with infinitely differentiable coefficients $a(x)$ and $b(x)$ decreasing as $|x|\to\infty$. For the case when the functions $a(x)$ and $b(x)$ can be expanded as $x\to\pm\infty$ on asymptotic series of the form \begin{gather*} a(x)=a_1|x|^{-\alpha_1}+\dots +a_i|x|^{-\alpha_i}+\dots , \\ b(x)=b_1|x|^{-\beta_1}+\dots +b_i|x|^{-\beta_i}+\dots , \end{gather*} where $\alpha_m$, $\beta_m\uparrow\infty$ as $m\to\infty$, $\alpha_1>1$, $\beta_1>2$, she constructs and justifies asymptotic expansion of the fundamental solution $G(x, s, t)$ to within any power of $G(x, s, t)$ uniformly with respect to all $x$ and $s$ in $\mathbf R^1$. Bibliography: 12 titles.

[1] Guschin A. K., Mikhailov V. P., “O stabilizatsii reshenii zadachi Koshi dlya parabolicheskogo uravneniya s odnoi prostranstvennoi peremennoi”, Tr. MMO, 112 (1971), 181–202 | Zbl

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

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

[4] Bagirov L. A., Shubin M. A., “O stabilizatsii resheniya zadachi Koshi dlya parabolicheskikh uravnenii s koeffitsientami, pochti periodicheskimi po prostranstvennym peremennym”, Differents. uravneniya, 11:12 (1975), 2205–2209 | MR | Zbl

[5] Ilin A. M., Khasminskii R. Z., “Asimptoticheskoe povedenie reshenii parabolicheskikh uravnenii i ergodicheskoe svoistvo neodnorodnykh diffuzionnykh protsessov”, Matem. sb., 60(102) (1963), 366–392

[6] Khasminskii R. Z., “Ergodicheskie svoistva vozvratnykh diffuzionnykh protsessov i stabilizatsiya reshenii zadachi Koshi dlya parabolicheskikh uravnenii”, Teoriya veroyatnostei i ee primeneniya, 5:2 (1960), 196–214

[7] 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

[8] Kozlov S. M., “Asimptotika na beskonechnosti fundamentalnykh reshenii uravnenii s pochti periodicheskimi koeffitsientami”, Vestn. MGU. Ser. 1. Matematika, mekhanika, 1980, no. 4, 11–16 | MR | Zbl

[9] Zhikov V. V., Kozlov S. M., Oleinik O. A., Kho Ten-Ngoan, “Usrednenie i $G$-skhodimost differentsialnykh operatorov”, UMN, 34:5 (1979), 65–133 | MR | Zbl

[10] Porper F. O., Eidelman S. D., “Dvustoronnie otsenki fundamentalnykh reshenii parabolicheskikh uravnenii vtorogo poryadka i nekotorye ikh prilozheniya”, UMN, 39:3 (1984), 107–156 | MR | Zbl

[11] Koddington E. A., Levinson N., Teoriya obyknovennykh differentsialnykh uravnenii, IL, M., 1958

[12] Marchenko V. A., Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR