  We introduce the concept of distillability sudden death, i.e., free entangled
states can evolve into non-distillable (bound entangled or separable) states in
finite time under local noise. We describe the phenomenon through a specific
model of local dephasing noise and compare the behavior of states in terms of
the Bures fidelity. Then we propose a few methods to avoid distillability
sudden death of states under (general) local dephasing noise, so that free
entangled states can be robust against decoherence. Moreover, we find that
bound entangled states are unstable in the limit of infinite time.
