an ill-posed problem
Some perfectly innocent-looking PDE questions are traps. They look like they should have an answer, but they fail one of Hadamard's three demands — and the most dangerous failure is the loss of stability, where a microscopic error in the data turns into a catastrophic error in the answer. An ill-posed problem is exactly this: a problem that lacks existence, or uniqueness, or (most often, and most insidiously) continuous dependence on the data. Because real data always carries small errors, an unstable problem is not just inconvenient — it is numerically and physically meaningless as stated.
Hadamard's own example makes the failure vivid. Take Laplace's equation u_xx + u_yy = 0 — an elliptic equation — and try to pose it as an evolution from Cauchy data on the line y = 0: u(x, 0) = 0 and u_y(x, 0) = (1/n) sin(n x). The data is tiny: as n grows it shrinks uniformly toward zero. But the solution that the equation forces is u(x, y) = (1/n^2) sin(n x) sinh(n y), and sinh(n y) grows like e^(n y). For any fixed height y > 0, that solution blows up explosively as n increases, even though its data was vanishing. A vanishing change in the data produces an unbounded change in the answer — stability is destroyed, so the Cauchy problem for Laplace's equation is ill-posed. The backward heat equation is the other textbook case: running diffusion in reverse, trying to recover a sharp past from a smoothed present, amplifies every high-frequency wiggle by e^(+ k n^2 t) and is hopelessly unstable.
The lesson is not that these equations are defective — Laplace's equation is beautifully well-posed with boundary data, and the heat equation is well-posed running forward. The lesson is that the type of the equation dictates the data: ask an elliptic equation to evolve, or a diffusion to run backward, and you have posed the wrong question. Ill-posed problems do arise in real life — deblurring an image, inverse scattering, reconstructing a source from distant measurements — and they are not abandoned but regularized: extra assumptions or penalty terms are added to restore stability and pull a sensible approximate answer out of an unstable question.
Hadamard's example: for Laplacian u = 0 the Cauchy data u(x,0)=0, u_y(x,0)=(1/n) sin(nx) shrinks to zero as n grows, yet forces u(x,y)=(1/n^2) sin(nx) sinh(ny), which grows like e^(ny)/n^2 at any height y>0. Vanishing data, exploding solution — no continuous dependence, so the problem is ill-posed.
The Cauchy problem for Laplace's equation: tiny data, unbounded solution. The textbook ill-posed problem.
Ill-posed does not mean unsolvable in practice: such problems are routinely tackled by regularization, which adds prior information or a stabilizing penalty to recover a meaningful approximate answer — but the raw, unregularized problem genuinely has no stable solution.