  From the dynamics of a brane-world with matter fields present in the bulk,
the bulk metric and the black string solution near the brane are generalized,
when both the dynamics of inhomogeneous dust/generalized dark radiation on the
brane-world and inhomogeneous dark radiation in the bulk as well are considered
-- as exact dynamical collapse solutions. Based on the analysis on the
inhomogeneous static exterior of a collapsing sphere of homogeneous dark
radiation on the brane, the associated black string warped horizon is studied,
as well as the 5D bulk metric near the brane. Moreover, the black string and
the bulk are shown to be more regular upon time evolution, for suitable values
for the dark radiation parameter in the model, by analyzing the physical soft
singularities.
