On the stabilization rate of solutions of the Cauchy problem for nondivergent parabolic equations with growing lower-order term

In the Cauchy problem
\begin{equation*} \begin{gathered} L_1u\equiv Lu+(b,\nabla u)+cu-u_t=0,\quad(x,t)\in D,\\ u(x,0)=u_0(x),\quad x\in\mathbb R^N, \end{gathered} \end{equation*}
for nondivergent parabolic equation with growing lower-order term in the half-space $\overline D=\mathbb R^N\times[0,\infty)$, $N\geqslant3$, we prove sufficient conditions for exponential stabilization rate of solution as $t\to+\infty$ uniformly with respect to $x$ on any compact $K$ in $\mathbb R^N$ with any bounded and continuous in $\mathbb R^N$ initial function $u_0(x)$.