  We study the star-formation activity in a sample of $\sim$ 56,000 brightest
cluster galaxies (BCGs) at $0.05 < z < 0.42$ using optical and infra-red data
from SDSS and WISE. We estimate stellar masses and star-formation rates (SFR)
through SED fitting and study the evolution of the SFR with redshift as well as
the effects of BCG stellar mass, cluster halo mass and cooling time on star
formation. Our BCGs have $SFR = 1.4\times10^{-3}-275.2$ [$\rm M_{\odot}$/yr]
and $sSFR=5 \times 10^{-15}-6 \times 10^{-10}$ [yr$^{-1}$] . We find that
star-forming BCGs are more abundant at higher redshifts and have higher $SFR$
than at lower redshifts. The fraction of star-forming BCGs ($f_{\rm SF}$)
varies from 30% to 80% at $0.05 < z < 0.42$. Despite the large values of
$f_{\rm SF}$, we show that only 13% of the BCGs lie on the star-forming main
sequence for field galaxies at the same redshifts. We also find that $f_{\rm
SF}$ depends only weakly on $\rm M_{200}$, while it sharply decreases with $\rm
M_{*}$. We finally find that the $SFR$ in BCGs decreases with increasing $\rm
t_{cool}$, suggesting that star formation is related to the cooling of the
intra-cluster medium. However, we also find a weak correlation of $\rm M_{*}$
and $\rm M_{200}$ with $\rm t_{cool}$, suggesting that AGN are heating the
intra-cluster gas around the BCGs. We compare our estimates of $SFR$ with the
predictions from empirical models for the evolution of the $SFR$ with redshift,
finding that the transition from a merger dominated to a cooling-dominated star
formation may happen at $z < 0.6$.
