Allen-Cahn equation

Question

Why is it that the Allen-Cahn equation is in some lecture notes written as $u_t = \Delta u - \epsilon^{-2} f(u)$ and in others $u_t = \epsilon \Delta u - \epsilon^{-1} f(u)$? The Allen-Cahn equation is the $L^2$-gradient flow of the Allen-Cahn energy, but this means that there are two different energies resulting in the two above eqautions. I don't understand how you can conclude that the two equations describe the same dynamics.