FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

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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Last modified: December 15, 1998