From Zachmanoglou and Thoe's "Intro to PDEs"

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

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