 Time reparametrization for a separable conservation-law flux

Suppose $a(t)>0$ for positive times. The strictly increasing new time $T(t)={\int }_0^{t}a(s)ds$ converts the scalar conservation law $u_t+a(t){\partial }_{x}F(u)=0$ into $U_T+{\partial }_{x}F(U)=0$, with $u(x,t)=U(x,T(t))$. The chain rule proves the claim for classical solutions; a change of variables in the weak formulation proves it for weak solutions and preserves the time orientation of the Kruzhkov entropy inequality. An isolated zero of $a$ at the initial endpoint does not obstruct the positive-time transformation.

The Rankine-Hugoniot condition becomes $d{x}_s/dt=a(t)[F(u)]/[u]$, while characteristic curves obey $dx/dt=a(t)F^{\prime }(u)$. Thus a straight shock in the new time may accelerate in physical time. For $a(t)=t$, a constant-speed trajectory in $T=t^2/2$ is a parabola in $(x,t)$. This separates genuine changes in wave structure from a common time-dependent speed factor.