= Time reparametrization for a separable conservation-law flux
{title2=$u_t+a(t)F(u)_x=0,\quad T=\int_0^ta(s)\,ds$}
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> $u_t+a(t)\partial_xF(u)=0$ into $U_T+\partial_xF(U)=0$, 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 $dx_s/dt=a(t)[F(u)]/[u]$, while <characteristic curves> obey $dx/dt=a(t)F'(u)$. 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.
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