On Homogenization of Periodic Parabolic Systems

We study homogenization in the small period limit for a periodic parabolic Cauchy problem in $\mathbb{R}^d$ and prove that the solutions converge in $L_2(\mathbb{R}^d)$ to the solution of the homogenized problem for each $t>0$. For the $L_2(\mathbb{R}^d)$-norm of the difference, we obtain an order-sharp estimate uniform with respect to the $L_2(\mathbb{R}^d)$-norm of the initial value.