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

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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. 
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     author = {M. A. Evgrafov},
     title = {Estimates of the fundamental solution of a~parabolic equation},
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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