  We study the gravitational collapse of a dust dark matter star in a
$\Lambda$-background. We consider two distinct cases: First we do not have a
dark matter and dark energy coupling; second, we consider that $\Lambda $ decay
in dark particles. The approach adopted assumes a modified matter expansion
rate and we have formation of a black hole, since that, we have the formation
of an apparent horizon. A brief comparison of the process of the matter
condensation using the gravitational collapse approach and the linear scalar
perturbation theory is considered.
