  We investigate the evolution and number distribution of radio halos in galaxy
clusters. Without re-acceleration 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 $\sim$ 0.1 Gyr. Radio halos could last for $\sim$ Gyr if a
significant level of re-acceleration 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
intra-cluster 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 $\sim$ Gyr. Our results indicate that a
significant level of re-acceleration is necessary for the observed radio halos
and the secondary electrons may not be a dominant origin for radio halos.
