Will black holes eventually engulf the Universe?

The Babichev-Dokuchaev-Eroshenko model for the accretion of dark energy onto black holes has been extended to deal with black holes with non-static metrics. The possibility that for an asymptotic observer a black hole with large mass will rapidly increase and eventually engulf the Universe at a finite time in the future has been studied by using reasonable values for astronomical parameters. It is concluded that such a phenomenon is forbidden for all black holes in quintessential cosmological models.