Heat equation in a periodic domain with special initial data

We consider the initial-boundary value problem with the Neumann boundary condition for the classical linear heat equation in unbounded domains $\Omega ⊊R^d$ which are periodic in all directions of the Cartesian coordinate system. Generalizing the results of a previous paper by the authors, we apply Floquet transform methods to obtain results on the large time decay rates of the solution in the sup-norm. We observe that for a general, integrable initial data, the solution decays at the same rate $t^{-d/2}$ as in the case of the Cauchy problem in the entire Euclidean space. We also consider special initial data with vanishing x-integral and obtain a faster decay rate. In the main results of the paper we pose for the initial data certain more detailed conditions, which are related to the lowest eigenvalue and eigenfunction of the model problem coming from the Floquet transform. Faster decay rates are obtained for such initial data.