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)^{k}D_{x}^{k}(a(x,
$$**y**
$)D$_{x}^{k}u) +
å _{i = l}^{m} å _{|a| =
|b|=i}
(-1)^{i}
$D$_{y}^{a} $(b$_{ab}(x, $$**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, |y_{1}| < 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(-κ_{2}t^{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 $k\; =\; l\; =\; m\; =\; 1$.

Location: http://mech.math.msu.su/~fpm/eng/k06/k064/k06408h.htm

Last modified: February 17, 2007