  We introduce a class of stochastic Allen-Cahn equations with a mobility
coefficient and colored noise. For initial data with finite free energy, we
analyze the corresponding Cauchy problem on the $d$-dimensional torus in the
time interval $[0,T]$. Assuming that $d\le 3$ and that the potential has
quartic growth, we prove existence and uniqueness of the solution as a process
$u$ in $L^2$ with continuous paths, satisfying almost surely the regularity
properties $u\in C([0,T]; H^1)$ and $u\in L^2([0,T];H^2)$.
