Geodesic
Estimates of the fundamental solution of a parabolic equation
Sbornik. Mathematics, Tome 40 (1981) no. 3, pp. 305-324
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper the behavior of the fundamental solution of the parabolic equation $$ \frac{\partial u}{\partial t}+P\biggl(\mathbf x,\frac1i\frac\partial{\partial\mathbf x}\biggr)u=0,\qquad\mathbf x\in\mathbf R^n,\quad t>0, $$ as $t\to+0$ uniformly with respect to $\mathbf x$ is investigated. The basic result is of the form $$ \varlimsup_{t\to+0}t^\frac1{2m-1}\ln|G(\mathbf x,\mathbf y,t)|\leqslant[\rho_P(\mathbf x,\mathbf y)]^\frac{2m}{2m-1}\cdot\sin\frac\pi{2(2m-1)}, $$ where $\rho_P(\mathbf x,\mathbf y)$ is the distance between $\mathbf x$ and $\mathbf y$ in a Finsler metric defined by the polynomial $P$. Bibliography: 4 titles. 
@article{SM_1981_40_3_a1,
     author = {M. A. Evgrafov},
     title = {Estimates of the fundamental solution of a~parabolic equation},
     journal = {Sbornik. Mathematics},
     pages = {305--324},
     year = {1981},
     volume = {40},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_40_3_a1/}
}
TY  - JOUR
AU  - M. A. Evgrafov
TI  - Estimates of the fundamental solution of a parabolic equation
JO  - Sbornik. Mathematics
PY  - 1981
SP  - 305
EP  - 324
VL  - 40
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1981_40_3_a1/
LA  - en
ID  - SM_1981_40_3_a1
ER  - 
%0 Journal Article
%A M. A. Evgrafov
%T Estimates of the fundamental solution of a parabolic equation
%J Sbornik. Mathematics
%D 1981
%P 305-324
%V 40
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1981_40_3_a1/
%G en
%F SM_1981_40_3_a1
M. A. Evgrafov. Estimates of the fundamental solution of a parabolic equation. Sbornik. Mathematics, Tome 40 (1981) no. 3, pp. 305-324. http://geodesic.mathdoc.fr/item/SM_1981_40_3_a1/

[1] M. A. Evgrafov, M. M. Postnikov, “Asimptotika funktsii Grina parabolicheskikh i ellipticheskikh uravnenii s postoyannymi koeffitsientami”, Matem. sb., 82(124) (1970), 3–29 | MR | Zbl

[2] S. R. S. Varadhan, “On the behavior of the fundamental solution of the heat equation with variable coefficients”, Comm. Pure Appl. Math., 20 (1967), 431–455 | DOI | MR | Zbl

[3] Y. Kannai, “Off diagonal short time asymptotics for fundamental solutions of diffusion equations”, Comm. partial differential equations, 2(8) (1977), 781–830 | DOI | MR

[4] M. A. Evgrafov, M. M. Postnikov, “Esche ob asimptotike funktsii Grina parabolicheskikh uravnenii s postoyannymi koeffitsientami”, Matem. sb., 92(134) (1973), 171–194 | MR | Zbl