Admit responsibility (5)
'admit responsibility' is the definition.
(I've seen this in another clue)
This is all the clue.
(Other definitions for own up that I've seen before include "Accept responsibility" , "be honest" , "Confess, admit guilt (3.2)" , "Admit to wrongdoing" , "Come clean (about wrongdoing)" .)
(Another definition for ownup that I've seen is " confess".)