Reduction on characteristics for continuous solutions of a scalar balance law

We consider continuous solutions $u$ to the balance equation \[ \partial_t u(t,x) + \partial_x [f(u(t,x))] = g(t,x) \quad f \in C^2(R), g \in L^\infty (R^+ \times R) \] for a bounded source term $g$. Continuity improves to Holder continuity when $f$ is uniformly convex, but it is not more regular in general. We discuss the reduction to ODEs on characteristics, mainly based on the joint works [5, 1]. We provide here local Lipschitz regularity results holding in the region where $f'(u) f''(u) \ne 0$ and only in the simpler case of autonomous sources $g = g(x)$, but for solutions $u(t,x)$ which may depend on time. This corresponds to a local Lipschitz regularity result, in that region, for the system of ODEs \[ \begin{cases} {\dot\gamma(t) = f'(u(t,\gamma(t))) \\ \frac{d}{dt} u(t,\gamma(t)) = g(gamma(t)) \end{cases} \]