Stabilization of a solution of the first mixed problem for a quasi-elliptic evolution equation

In a cylindrical domain $D=(0,\infty)\times\Omega$, where $\Omega$ is an unbounded subdomain of $\mathbb R_{n+1}$, one considers the first mixed problem for a higher order equation
\begin{gather*} u_t+Lu=0, \\ Lu\equiv\sum_{i=q}^k(-1)^iD_x^i(a_i(x,{\mathbf y})D_x^iu)+ \sum_{i=l}^m\,\sum_{|\alpha|=|\beta|=i}(-1)^i D_{\mathbf y}^\alpha(b_{\alpha\beta}(x,{\mathbf y})D_{\mathbf y}^\beta u), \\ q\leqslant k,\quad l\leqslant m,\quad q,k,l,m\in\mathbb N,\quad x\in\mathbb R,\quad \mathbf y\in\mathbb R_n, \end{gather*}
with homogeneous boundary conditions and compactly supported initial function. A new method of obtaining an upper estimate of the $L_2$-norm $\|u(t)\|$ of the solution of this problem is put forward, which works in a broad class of domains and equations. In particular, in domains $\{(x,{\mathbf y})\in\mathbb R_{n+1}:|y_1|<x^a\}$, $0<a<q/l$, for the operator $L$ with symbol satisfying a certain condition this estimate takes the following form:
$$ \|u(t)\|\leqslant M\exp(-\kappa_2t^b)\|\varphi\|,\qquad b=\frac{q-{la}}{q-{la}+2laq}\,. $$
The estimate is shown to be sharp in a broad class of unbounded domains for $q=k=l=m=1$, that is, for second-order parabolic equations.