  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).
