  In this work we study the evolution of matter-density perturbations for an
arbitrary $\Lambda(t)$ model, and specialize our analysis to the particular
phenomenological law $\Lambda = \Lambda_0 + 3\beta H^2$. We study the evolution
of the cosmic star formation rate in this particular dark energy scenario and,
by constraining the $\beta$ parameter using both the age of the universe and
the cosmic star formation rate curve, we show that it leads to a reasonable
physical model for $\beta\lesssim 0.1$.
