  We study the dynamics of ordering in ferromagnets via Monte Carlo simulations
of the Ising model, employing the Glauber spin-flip mechanism, in space
dimensions $d=2$ and $3$. Results for the persistence probability and the
domain growth are discussed for quenches to various temperatures ($T_f$) below
the critical one ($T_{c}$), from different initial temperatures $T_{i} \geq
T_{c}$. In long time limit, for $T_{i} > T_{c}$, the persistence probability
exhibits power-law decay with exponents $\theta \simeq 0.22$ and $\simeq 0.18$
in $d=2$ and $3$, respectively. For finite $T_i$, the early time behavior is a
different power-law whose life-time diverges and exponent decreases as $T_{i}
\rightarrow T_{c}$. The crossover length between the two steps diverges as the
equilibrium correlation length. $T_i=T_c$ is expected to provide a {\it{new
universality class}} for which we obtain $\theta \simeq 0.035$ in $d=2$ and
$\simeq 0.10$ in $d=3$. The time dependence of the average domain size $\ell$,
however, is observed to be rather insensitive to the choice of $T_i$.
