FPM 2006, vol. 12, no. 4, pp. 113-132

FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 4, PAGES 113-132

Decay of the solution of the first mixed problem for a high-order parabolic equation with minor terms

L. M. Kozhevnikova
F. Kh. Mukminov

Abstract

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In a cylindric domain $D =
(0,$ ¥) ´ W, where W Ì R_n+1 is an unbounded domain, the first mixed problem for a high-order parabolic equation

u

_t + (-1)^kD_x^k(a(x,

)D

_x^ku) + å _{i = l}^m å _{|a| =
|b|=i} (-1)ⁱ

D

_y^a

(b

_ab(x,

y)D_y^b

u) = 0,

l £ m, k, l, m Î

N, is considered. The boundary values are homogeneous and the initial value is a finite function. In terms of new geometrical characteristic of domain, the upper estimate of

L

₂-norm

||u(t)||

of the solution to the problem is established. In particular, in domains

{(x,

y) Î R_n+1 | x > 0, |y₁| < x^a},

0 < a
< q/l

, under the assumption that the upper an lower symbols of the operator

L

are separated from zero, this estimate takes the form

||u(t)||

£ M exp(-κ₂t^b) ||f||, b = (k - la)/(k - la + 2lak).

This estimate is determined by minor terms of the equation. The sharpness of the estimate for the wide class of unbounded domains is proved in the case $k = l = m = 1$ .

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