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

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 $\nu(\eta)\in C^1(\mathbf R^n)$.
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