  Here we generalized a previous model of gravastar consisted of an internal de
Sitter spacetime, a dynamical infinitely thin shell with an equation of state,
but now we consider an external de Sitter-Schwarzschild spacetime. We have
shown explicitly that the final output can be a black hole, a "bounded
excursion" stable gravastar, a stable gravastar, or a de Sitter spacetime,
depending on the total mass of the system, the cosmological constants, the
equation of state of the thin shell and the initial position of the dynamical
shell. We have found that the exterior cosmological constant imposes a limit to
the gravastar formation, i.e., the exterior cosmological constant must be
smaller than the interior cosmological constant. Besides, we have also shown
that, in the particular case where the Schwarzschild mass vanishes, no stable
gravastar can be formed, but we still have formation of black hole.
