Large time behavior in $\mathbb{R}^N$ for linear parabolic equations with periodic coefficients

In this paper we study the asymptotic behavior of solutions of linear parabolic equations in $\mathbb{R}^N$ with periodic coefficients and $L^1$ initial data, as $t\rightarrow \infty$ . It was already known that, in a first approximation, solutions behave as the fundamental solution of the homogenized system. We use the Bloch waves decomposition to obtain a complete expansion of the solutions as $t\rightarrow \infty$ .