We show that for any attractive Glauber-Exclusion process on the one-dimensional lattice of size N with periodic boundary condition, if the corresponding hydrodynamic limit equation has a reaction term with a strictly convex potential, then the total-variation mixing time is of order O(log N). In particular, the result covers the full high-temperature regime in the original model introduced by De Masi, Ferrari and Lebowitz (1985).
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