Stabilization of solutions of pseudo-differential parabolic equations in unbounded domains

We consider the first mixed problem in a cylindrical domain $D=(0,\infty)\times\Omega$ for a pseudo-differential parabolic equation with homogeneous Dirichlet boundary conditions and a finitely supported initial function. We find upper bounds for the $L_2$-norm of a solution as $t\to\infty$ in terms of a geometric characteristic introduced earlier by the author for an unbounded domain $\Omega\subset\mathbb R^n$, $n\geqslant 2$, in the case of a higher-order parabolic equation.