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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 thebrane-world and inhomogeneous dark radiation in the bulk as well are considered-- as exact dynamical collapse solutions. Based on the analysis on theinhomogeneous static exterior of a collapsing sphere of homogeneous darkradiation 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 andthe bulk are shown to be more regular upon time evolution, for suitable valuesfor the dark radiation parameter in the model, by analyzing the physical softsingularities.

Regular Bulk Solutions in Brane-worlds with Inhomogeneous Dust and Generalized Dark Radiation