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?