Estimates of the fundamental solution of a~parabolic equation

M. A. Evgrafov

Sbornik. Mathematics, Tome 40 (1981) no. 3, pp. 305-324

Résumé

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.