  We study the time-dependence of the local persistence probability during a
non-stationary time evolution in the disordered contact process in $d=1,2$, and
$3$ dimensions. We present a method for calculating the persistence with the
strong-disorder renormalization group (SDRG) technique, which we then apply in
the critical point analytically for $d=1$ and numerically for $d=2,3$.
According to the results, the average persistence decays at late times as an
inverse power of the logarithm of time, with a universal, dimension-dependent
generalized exponent. For $d=1$, the distribution of sample-dependent local
persistences is shown to be characterized by a universal limit distribution of
effective persistence exponents. By a phenomenological approach of rare-region
effects in the active phase, we obtain a non-universal algebraic decay of the
average persistence for $d=1$, and enhanced power laws for $d>1$. As an
exception, for randomly diluted lattices, the algebraic decay holds to be valid
for $d>1$, which is explained by the contribution of dangling ends. Results on
the time-dependence of average persistence are confirmed by Monte Carlo
simulations. We also prove the equivalence of the persistence with a return
probability, a valuable tool for the argumentations.
