A set of gradient stresses is used in Landau thermodynamic theory to explain the dielectric diffusion of BaTiO3 films grown on thick copper foils. Every grain in the films is treated as a single domain core that is surrounded by boundaries with low dielectric constant. The dielectric diffusion is mainly induced by the diffusive phase transition caused by the gradient stresses. The low dielectric constant boundaries suppress the peak value of the dielectric constant. The results agree with the experiments.