On the stabilization rate of solutions of the Cauchy problem for a parabolic equation with lower-order terms

For a parabolic equation in the half-space $\overline D=\mathbb R^N\times[0,\infty)$, $N\geqslant3$, we consider the Cauchy problem
\begin{gather*} L_1u\equiv Lu+c(x,t)u-u_t=0,\quad (x,t)\in D,\\ u(x,0)=u_0(x),\quad x\in\mathbb R^N. \end{gather*}
Depending on estimates on the coefficient $c(x,t),$ we establish power or exponential rate of stabilization of solutions of the Cauchy problem равномерно по $x$ на каждом компакте $K$ в $\mathbb R^N$ для произвольной ограниченной непрерывной в $\mathbb R^N$ начальной функции $u_0(x)$.