Solution
= Solution
The compact support just established and $f(0)=0$ permit differentiation of the <antiderivative> and integration of the <scalar conservation law> from $-\infty$:
$$
U_x=u,\qquad U_t=-f(u),\qquad \boxed{U_t+f(U_x)=0,\quad U(0,x)=\int_{-\infty}^x u_0(y)\,dy.}
$$
This is a <Hamilton-Jacobi equation>. There is no arbitrary function of time: the normalization at $-\infty$ and the zero exterior flux determine it.