Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 5 / 1 / 3 / c / Solution

The compact support just established and $f(0)=0$ permit differentiation of the antiderivative and integration of the scalar conservation law from $-\infty$:This is a Hamilton-Jacobi equation. There is no arbitrary function of time: the normalization at $-\infty$ and the zero exterior flux determine it.