  We provide a complete characterization of the evolution of entanglement
between two oscillators coupled to a common environment. For initial Gaussian
states we identify three phases with different qualitative long time behavior:
There is a phase where entanglement undergoes a sudden death (SD). Another
phase (SDR) is characterized by an infinite sequence of events of sudden death
and revival of entanglement. In the third phase (NSD) there is no sudden death
of entanglement, which persist for long time. The phase diagram is described
and analytic expressions for the boundary between phases are obtained.
Numerical simulations show the accuracy of the analytic expressions. These
results are applicable to a large variety of non--Markovian environments. The
case of non--resonant oscillators is also numerically investigated.
