Constraints on $\Lambda(t)$-cosmology with power law interacting dark
  sectors
  Motivated by the cosmological constant and the coincidence problems, we
consider a cosmological model where the cosmological constant $\Lambda_0$ is
replaced by a cosmological term $\Lambda(t)$ which is allowed to vary in time.
More specifically, we are considering that this dark energy term interacts with
dark matter through the phenomenological decay law
$\dot{\rho}_{\Lambda}=-Q\rho_{\Lambda}^{n}$. We have constrained the model for
the range $n\in[0,10]$ using various observational data (SNeIa, GRB, CMB, BAO,
OHD), emphasizing the case where $n=3/2$. This case is the only one where the
late-time value for the ratio of dark energy density and matter energy density
$\rho_{\Lambda}/\rho_{m}$ is constant, which could provide an interesting
explanation to the coincidence problem. We obtain strong limits on the model
parameters which however exclude the region where the coincidence or the
cosmological constant problems are significantly ameliorated.
