We investigate the evolution and number distribution of radio halos in galaxy clusters. Without reacceleration or regeneration, the relativistic electrons responsible for the diffuse radio emission will lose their energy via inverse Compton and synchrotron losses in a rather short time, and radio halos will have lifetimes ~0.1 Gyr. Radio halos could last for ~Gyr if a significant level of reacceleration is involved. The lifetimes of radio halos would be comparable with the cosmological time if the radio-emitting electrons are mainly the secondary electrons generated by pion decay following proton-proton collisions between cosmic-ray protons and the thermal intracluster medium within the galaxy clusters. Adopting both observational and theoretical constraints for the formation of radio halos, we calculate the formation rates and the comoving number density of radio halos in the hierarchical clustering scheme. Comparing with observations, we find that the lifetimes of radio halos are ~Gyr. Our results indicate that a significant level of reacceleration is necessary for the observed radio halos and the secondary electrons may not be a dominant origin for radio halos.
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