  We analyze the evolution of the Gaussian discord between two resonant
harmonic oscillators coupled to a common environment. For this, we use the same
tools we applied before to fully characterize the evolution of the entanglement
in this system [ Phys. Rev. Lett. 100 220401 (2008)]. The asymptotic value of
Gaussian discord is obtained as a function of parameters characterizing the
environment (temperature, couplings, etc.) and the initial state of the system
(initial squeezing, initial purity, etc.) The type of Gaussian measurement
optimizing the extraction of information between the oscillators is fully
characterized by means of a phase diagram. Such a diagram (with phases
corresponding to homodyne or heterodyne measurements) has similar topology to
the one describing dynamical phases for the entanglement. We present evidence
pointing to the fact that Gaussian discord is not always a good approximation
of true discord as the asymptotic value of the former is shown to be a
nondecreasing function of temperature (in the high-temperature regime),
reaching an asymptotic value of log(2) for a pure initial state (and lower
values for mixed initial states).
