Having a PDE plus conditions is not yet a guarantee of a sensible problem. Jacques Hadamard distilled what "sensible" means into three requirements. A problem is well-posed if a solution

• exists,
• is unique, and
• depends continuously on the data — small changes in the inputs cause only small changes in the solution (stability).

Miss any one and the problem is ill-posed. The third is the subtle one, and the one numerical computation cares about most.

Why stability matters

Real data — measurements, initial profiles — always carry small errors. If a tiny perturbation could explode into a wildly different solution, no computed answer could be trusted. Stability is what makes a model predictive.

The classic ill-posed example is the backward heat equation: run diffusion in reverse to reconstruct the past temperature from the present. Forward diffusion smooths bumps away; running it backward must amplify them, and the higher the frequency the more violently it grows. A mode \sin(nx) that decayed like e^{-\alpha n^2 t} forward must grow like e^{+\alpha n^2 t} backward — an unbounded blow-up from the faintest high-frequency noise. Existence and uniqueness can hold, yet the lack of stability makes the problem useless.