Nonexistence of Global Solutions to the Nonhomogeneous Wave Equation, Regardless of Boundary Conditions

The fact that the ordinary derivative y′ is known to be a Darboux function implies that the right‐hand side of the differential equation y′ = f(x, y) must also be such a function. Since it is now known that the mixed hyperbolic derivative uxy is also a Darboux function, this implies that under certain mild conditions the equation uxx − utt = f may not have a classical global solution unless f has the proper Darboux structure. As with the ordinary differential equation, this nonexistence result does not depend on boundary conditions.