  This paper considers a family of second-order periodic parabolic equations
with highly oscillating potentials, which have been considered many times for
the time-varying potentials in stochastic homogenization. Following a standard
two-scale expansions illusion, we can guess and succeed in determining the
homogenized equation in different cases that the potentials satisfy the
corresponding assumptions, based on suitable uniform estimates of the
$L^2(0,T;H^1(\Omega))$-norm for the solutions. To handle the more singular case
and obtain the convergence rates in $L^\infty(0,T;L^2(\Omega))$, we need to
estimate the Hessian term as well as the t-derivative term more exactly, which
may be depend on $\varepsilon$. The difficulty is to find suitable uniform
estimates for the $L^2(0,T;H^1(\Omega))$-norm and suitable estimates for the
higher order derivative terms.

Analysis of PDEs