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

In a cylindric domain $D=(0,\infty)\times\Omega$, where $\Omega\subset \mathbb{R}_{n+1}$ is an unbounded domain, the first mixed problem for a high-order parabolic equation
\begin{gather*} u_t+(-1)^kD_x^k(a(x,\mathbf{y})D_x^ku)+\sum\limits_{i=l}^m\sum\limits_{|\alpha|=|\beta|=i}(-1)^i D_\mathbf{y}^{\alpha}(b_{\alpha\beta}(x,\mathbf{y})D_{\mathbf{y}}^{\beta}u)=0, \\ l\leq m,\quad k,l,m\in \mathbb{N}, \end{gather*}
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 $\{(x,\mathbf y)\in\mathbb{R}_{n+1}\mid x>0,\ |y_1|<x^a\}$, $0<a<q/l$, under the assumption that the upper an lower symbols of the operator $L$ are separated from zero, this estimate takes the form
$$ \|u(t)\|\leq M\exp(-\varkappa_2t^{b})\|\varphi\|,\quad b=\frac{k-la}{k-la+2lak}. $$
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 $k=l=m=1$.