Asymptotics of the fundamental solution of a~Petrovskii parabolic equation with constant coefficients

S. G. Gindikin ; M. V. Fedoryuk

Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 519-542

Résumé

Let $P(\zeta)$, $\zeta\in\mathbf C^n$, be a homogeneous, parabolic polynomial of degree $2m$. Properties of the function $$ \nu(\eta)=\min_{\xi\in\mathbf R^n}\operatorname{Re}P(\xi+i\eta),\qquad\eta\in\mathbf R^n, $$ are investigated. Two-sided estimates are obtained for the fundamental solution $G(t,x)$ of the equation $$ \frac{\partial u}{\partial t}+P\biggl(\frac1i\frac\partial{\partial x}\biggr)u=0, $$ and an asymptotic decomposition is determined for $G(t,x)$ as $|x|^{2m}/t\to+\infty$ under the assumption that $\nu(\eta)\in C^1(\mathbf R^n)$. Bibliography: 14 titles.