Homogenization of the Parabolic Cauchy Problem in the Sobolev Class $H^1(\mathbb{R}^d)$

Homogenization in the small period limit for the solution $\mathbf{u}_\varepsilon$ of the Cauchy problem for a parabolic equation in $\mathbb{R}^d$ is studied. The coefficients are assumed to be periodic in $\mathbb{R}^d$ with respect to the lattice $\varepsilon\Gamma$. As $\varepsilon\to 0$, the solution $\mathbf{u}_\varepsilon$ converges in $L_2(\mathbb{R}^d)$ to the solution $\mathbf{u}_0$ of the effective problem with constant coefficients. The solution $\mathbf{u}_\varepsilon$ is approximated in the norm of the Sobolev space $H^1(\mathbb{R}^d)$ with error $O(\varepsilon)$; this approximation is uniform with respect to the $L_2$-norm of the initial data and contains a corrector term of order $\varepsilon$. The dependence of the constant in the error estimate on time $\tau$ is given. Also, an approximation in $H^1(\mathbb{R}^d)$ for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.