  The voter model with memory-dependent dynamics is theoretically and
numerically studied at the mean-field level. The `internal age', or time an
individual spends holding the same state, is added to the set of binary states
of the population, such that the probability of changing state (or activation
probability $p_i$) depends on this age. A closed set of integro-differential
equations describing the time evolution of the fraction of individuals with a
given state and age is derived, and from it analytical results are obtained
characterizing the behavior of the system close to the absorbing states. In
general, different age-dependent activation probabilities have different
effects on the dynamics. When the activation probability $p_i$ is an increasing
function of the age $i$, the system reaches a steady state with coexistence of
opinions. In the case of aging, with $p_i$ being a decreasing function, either
the system reaches consensus or it gets trapped in a frozen state, depending on
the value of $p_\infty$ (zero or not) and the velocity of $p_i$ approaching
$p_\infty$. Moreover, when the system reaches consensus, the time ordering of
the system can be exponential ($p_\infty>0$) or power-law like ($p_\infty=0$).
Exact conditions for having one or another behavior, together with the
equations and explicit expressions for the exponents, are provided.
