Reduction on characteristics for continuous solution of a scalar balance law

We consider continuous solutions u to the balance equations for a bounded source term. Continuity improves to Holder continuity when the flux f is uniformly convex, but it is not more regular in general. We discuss the reduction to ODEs on characteristics, strongly applying related results in longer works. We provide here an original local Lipschitz regularity results holding in the region where the product f'(u) times f''(u) is non vanishing---only in the simpler case of sources which are autonomous, but for solutions which may depend on time. This result relies on the deeper results of related works. This corresponds to a local Lipschitz regularity result, in that region, for the corresponding system of ODEs.