On the asymptotics of the fundamental solution of a high order parabolic equation

The behavior as $t\to\infty$ of the fundamental solution $G(x,s,t)$ of the Cauchy problem for the equation $u_t=(-1)^nu^{2n}_x+a(x)u$, $x\in\mathbb R^1$, $t>0$, $n>1$ is studied. It is assumed that the coefficient $a(x)\in C^{\infty}(\mathbb R^1)$ and as $x\to\infty$ expand into asymptotic series of the form
$$ a(x)=\sum_{j=0}^{\infty} a_{2n+j}^{\pm}x^{-2n-j}, \quad x\to\pm\infty. $$
The asymptotic expansion of the $G(x,s,t)$ as $t\to\infty$ is constructed and establiched for all $x,s\in\mathbb R^1$. The fundamental solution decays like power, and the decay rate is determined by the quantities of “principal” coefficients $a_{2n}^{\pm}$.