From Zachmanoglou and Thoe's "Intro to PDEs"

Consider the initial value problem for the equation with the initial curve the parabola . Show that unless the initial data satisfy a certain condition, the IVP has no global solution. However, if is any point on the curve different from , show that the IVP always has a solution in a neighborhood of . Is this true for ?

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