On the optimal regularizing effect for scalar balance laws with time-dependent source

We study the optimal regularizing effect for scalar balance laws with strictly convex fluxes and time-dependent nonlinear source terms. In a recent work, we proved that a suitable transversality condition between the source term and the degeneracy set of the flux restores full BVloc regularity. In this note, we show that this condition is optimal. If the flux has polynomial degeneracy and the source vanishes along a degenerate state on a nontrivial time interval, then the full BVloc regularizing effect cannot be expected in general. We construct an entropy solution showing that the fractional threshold BVloc1/p, already known from the regularizing theory, cannot be improved. The same construction also yields the corresponding sharp obstruction in the Sobolev scale.