the non-characteristic condition
Having decided to give a first-order PDE its data along a curve, you cannot choose that curve carelessly. The non-characteristic condition is the rule that keeps the choice honest: the data curve must cross the characteristics, not run alongside one. Cross them, and the data spreads to a unique solution; lie along one, and the problem falls apart.
Why? The characteristics are the channels that carry the solution outward from the data. If your data curve cuts across them — meeting each characteristic at a distinct point at a genuine angle — then every characteristic picks up exactly one starting value and carries it off, and the surface is determined. But if the data curve coincides with a characteristic, you have specified values along a single carrying channel, which constrains nothing in the neighbouring channels: either your data happens to agree with what the equation already forces along that characteristic (then infinitely many solutions fit) or it disagrees (then none do). Precisely, for a u_x + b u_y = c with data curve (x(r), y(r)), the condition is that the curve's tangent (dx/dr, dy/dr) is not parallel to the characteristic direction (a, b); algebraically, a (dy/dr) - b (dx/dr) is not zero. This is a transversality — a crossing — requirement.
The non-characteristic condition is the gatekeeper of the well-posed Cauchy problem. When it holds at a point of the data curve, the implicit function theorem guarantees a unique smooth solution in a neighbourhood. When it fails, you are warned: do not expect existence-and-uniqueness there. The same idea generalizes: for higher-order equations one asks that the data surface be non-characteristic for the principal symbol, the requirement behind the Cauchy-Kovalevskaya theorem.
For the transport equation u_t + c u_x = 0, the characteristics are the lines x - c t = const. Giving data on t = 0 is non-characteristic (the t=0 line crosses every characteristic once) — good. Giving data along a line x = c t + b, which is itself a characteristic, is characteristic data — bad, and the Cauchy problem there is not well-posed.
Cross the characteristics and the problem is well-posed; align with one and it is not.
The condition is local: a data curve can be non-characteristic in some places and tangent to a characteristic in others. The guarantee of a unique smooth solution holds only near the points where the crossing is genuine, and only until characteristics later cross each other.