= Expanding-flow transformation of a wave equation
{title2=$X=\rho e^{-t},\quad S=e^{-t}$}
The change of variables $X=\rho e^{-t}$, $S=e^{-t}>0$ turns
$$
u_{tt}-\partial_\rho((1-\rho^2)u_\rho)+\partial_\rho(\rho u_t)+\rho u_{\rho t}=0
$$
into $w_{SS}-w_{XX}=0$. Indeed, with $D=\partial_t+\rho\partial_\rho$, the original operator is $D^2+D-\partial_\rho^2$, while $D=-S\partial_S$ and $\partial_\rho=S\partial_X$. Its <characteristic curves> are $\rho=-1+Ce^t$ and $\rho=1+Ce^t$, including the characteristic lines $\rho=\pm1$. General smooth solutions have the form $F(e^{-t}(\rho+1))+G(e^{-t}(\rho-1))$. Zero <Cauchy data> on $-1<\rho<1$ at $t=0$ force vanishing throughout that strip for $t\geq0$, but not for all negative time: $u=(e^{-t}(\rho+1)-2)_+^3$ is a global $C^2$ counterexample. Thus the characteristic barriers describe a forward domain of dependence, not a barrier in both time directions.
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