Dark matter interacts with variable vacuum energy

We investigate a spatially flat Friedmann-Robertson-Walker (FRW) scenario with two interacting components, dark matter and variable vacuum energy (VVE) densities, plus two decoupled components, one is a baryon term while the other behaves as a radiation component. We consider a linear interaction in the derivative dark component density. We apply the $\chi^2$ method to the observational Hubble data for constraining the cosmological parameters and analyze the amount of dark energy in the radiation era for the model. It turns out that our model fulfills the severe bound of $\Omega_{x}(z\simeq 1100)<0.009$ at $2\sigma$ level, so is consistent with the recent analysis that includes cosmic microwave background anisotropy measurements from Planck survey, the future constraints achievable by Euclid and CMBPol experiments, reported for the behavior of the dark energy at early times, and fulfills the stringent bound $\Omega_{x}(z\simeq 10^{10})<0.04$ at $2\sigma$ level in the big-bang nucleosynthesis epoch. We also examine the cosmic age problem at high redshift associated with the old quasar APM 08279+5255 and estimate the age of the universe today.