In this talk, we consider the two-material optimal design problem for the time-averaged duality pairing between a heat source and the weak solution of an initial-boundary value problem for the heat equation with a two-material diffusion coefficient, under a volume constraint. In the elliptic (i.e., time-independent) case, this problem can be relaxed through homogenization, and the relaxation problem has a self-adjoint structure. In this elliptic setting, a direct relaxation with respect to the density is possible, and the convexity of the relaxed problem allows us to construct an optimal density. On the other hand, in the parabolic case, such a structure is in general not ensured due to the time-dependence. In this talk, we discuss the direct relaxability and the long-time behavior for the parabolic case in some specific settings.