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

In the paper, we study the sufficient conditions for the lower-order coefficient of the parabolic equation
$$ \Delta u+c(x,t)u-u_t=0\ \ \text{for}\ \ x\in\mathbb R^N,\ \ t>0, $$
under which its solution satisfying the initial condition
$$ u|_{t=0}=u_0(x)\ \ \text{for}\ \ x\in \mathbb R^N, $$
stabilizes to zero, i.e., there exists the limit
$$ \lim_{t\to\infty}{u(x,t)}=0, $$
uniform in $x$ from every compact set $K$ in $\mathbb R^N$ for any function $u_0(x)$ belonging to a certain uniqueness class of the problem considered and growing not rapidly than $e^{a|x|^b}$ with $a>0$ and $b>0$ at infinity.