In "The Yang Mills flow in four dimensions", M. Struwe proves that this flow converges, up to bubbling phenomena. And he has conjectured that this explosion in finite time should happen as proven for the harmonic heat flow by Chang, Ding and Ye. Looking on MathSciNet, I have found no proof or disproof of this fact. Is this question still open?
On the other hand, Donaldson proves that if the underlying bundle is stable there is a global existence. Has this hypothesis been weakened?