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 Rn+1 is an unbounded domain, the first mixed problem for a high-order parabolic equation

ut + (-1)kDxk(a(x, y )Dxku) + å i = lm å |a| = |b|=i (-1)i Dya (bab(x, y)Dyb 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 L2-norm ||u(t)|| of the solution to the problem is established. In particular, in domains {(x,y) Î Rn+1 | x > 0, |y1| < xa}, 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(-κ2tb) ||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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