  We consider the superposition of symmetric simple exclusion dynamics
speeded-up in time, with spin-flip dynamics in a one-dimensional interval with
periodic boundary conditions. We show that the mixing time has an exponential
lower bound in the system size if the potential of the hydrodynamic equation
has more than two local minima. We also apply our estimates to show that the
normalized hitting times of rare events converge to a mean one exponential
random variable if the potential has a unique minimum.
