(FUNDAMENTAL AND APPLIED MATHEMATICS)

1998, VOLUME 4, NUMBER 3, PAGES 1009-1027

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

E. F. Lelikova

Abstract

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The behavior as $t \to \infty$ of the fundamental solution $G\left(x,s,t\right)$ of the Cauchy problem for the equation $u$t=(-1)nu2nx+a(x)u, $x \in R1$, $t > 0$, $n > 1$ is studied. It is assumed that the coefficient $a\left(x\right) \in C\infty \left(R1\right)$ and as $x \to \infty$ expand into asymptotic series of the form

$a\left(x\right) = \sum _\left\{j=0\right\}^\left\{\infty \right\} a_\left\{2n + j\right\}^\left\{\pm \right\}x^\left\{-2n - j\right\}, \quad x \to \pm \infty .$

The asymptotic expansion of the $G\left(x,s,t\right)$ as $t \to \infty$ is constructed and establiched for all $x,s \in R1$. The fundamental solution decays like power, and the decay rate is determined by the quantities of "principal" coefficients $a_\left\{2n\right\}^\left\{\pm \right\}$.

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