FPM 1998, vol. 4, no. 3, pp. 1009-1027

FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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
$G(x,s,t)$ of  the  Cauchy  problem  for  the  equation  $u_t  = (- 1)^n
u^{2n}_x + a(x)u$,  $x \in \mathbb R^1$,  $t > 0$, $n > 1$ 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 $a_{2n}^{\pm}$.

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