We discuss a model for the autocatalytic reaction A + B → 2A on substrates where the reactants perform a compact exploration of the space, i.e., on lattices whose spectral dimensiond is < 2. For finite systems, the total time τ for the reaction to end scales according to two different regimes, for high and low concentrations of reactants. The functional dependence of τ on the volume of the substrate and the concentration of reactants is discussed within a mean-field approximation. Possible applications are discussed.