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Extensive Monte Carlo simulations are performed in order to evaluate both thelocal ($\theta_{l}$) and global ($\theta_{g}$) persistence exponents in theZiff-Gulari-Barshad (ZGB) (Phy. Rev. Lett. {\bf 56}, 2553, (1986)) irreversiblereaction model. In two dimensions and for the second-order irreversible phasetransition (IPT) we find, that both the local and the global persistenceexhibit power-law behavior with a crossover between two different time regimes.On the other hand, at the first-order IPT, characteristic of the ZGB, activesites are short lived and the persistence decays more abruptly, not being clearwhether it shows power law behavior or not. In order to analyze universalityissues, we have also studied another model with absorbing states, the contactprocess, and evaluated the local persistence exponent in dimensions from 1 to4. A striking apparent {\it super-universality} is reported: the localpersistence exponent seems to coincide in both one and two dimensional systems.Comment: Submitted PRE. 8 pages. (epsfig Style required

Numerical study of persistence in models with absorbing states