## DEs Proof Problem

From Zachmanoglou and Thoe's "Intro to PDEs"

Consider the initial value problem for the equation $D_1u=0$ with the initial curve the parabola $y=x^2$. Show that unless the initial data satisfy a certain condition, the IVP has no global solution. However, if $P$ is any point on the curve different from $(0, 0)$, show that the IVP always has a solution in a neighborhood of $P$. Is this true for $P=(0, 0)$?

Just a problem I found interesting but couldn't figure out. Guidance please?