Characteristic coordinate

A characteristic coordinate is a function constant along one family of characteristics. For a second-order equation in two variables, two independent characteristic coordinates reduce the principal part to a mixed derivative.

The change of variables $X=\rho e^{-t}$, $S=e^{-t}>0$ turnsinto $w_{SS}-w_{XX}=0$. 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 $\rho =-1+C{e}^t$ and $\rho =1+C{e}^t$, including the characteristic lines $\rho =\pm 1$. 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 $t\ge 0$, 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.