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 $t →
∞$ of the fundamental solution $G(x,s,t)$ of the Cauchy problem for the equation $u$ _t=(-1)ⁿu²ⁿ_x+a(x)u, $x ∈ R$ ¹ , $t >
0$ , $n >
1$ is studied. It is assumed that the coefficient $a(x) ∈ C$ ^∞ (R¹) and as $x
→ ∞$ 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 $t → ∞$ is constructed and establiched for all $x,s ∈ R$ 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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