a well-posed problem
/ Hadamard -> ah-dah-MAR /
When you set up a PDE to model something real — heat in a bar, a vibrating string, the shape of a soap film — you are quietly hoping for three things. You hope there is an answer at all. You hope there is only one, so the model makes a definite prediction. And you hope that if you measure your starting data slightly wrong, the answer is only slightly off, not wildly different — because every real measurement has error. Jacques Hadamard turned that hope into a checklist, and a problem that passes all three checks is called well-posed: it is a question worth asking, because nature could actually answer it.
Precisely, Hadamard's three conditions are: existence — a solution exists for the given data; uniqueness — there is exactly one such solution; and stability, or continuous dependence on the data — the solution changes continuously as the data changes, so small changes in the initial or boundary data produce only small changes in the answer. All three matter. Without existence, the model is empty; without uniqueness, it predicts nothing definite; without stability, it is useless in practice, because the unavoidable tiny errors in real data would be amplified into huge errors in the answer. The standard tool for proving uniqueness and stability together is an energy estimate: multiply the equation by u, integrate over the domain, and use integration by parts to bound a positive 'energy' of the solution by the data — if the data is zero the energy is zero, so the solution is zero, which gives uniqueness, and the same bound gives stability.
Well-posedness is the organizing question of the whole subject, not a footnote. The reason the classification elliptic/parabolic/hyperbolic is fundamental rather than cosmetic is precisely that it dictates which data make the problem well-posed: Cauchy/initial data for hyperbolic and parabolic equations, boundary data for elliptic equations. Pose the wrong kind of data — Cauchy data for the elliptic Laplace equation, or the heat equation run backward in time — and the very same equation becomes ill-posed, with solutions that either fail to exist or explode under the slightest perturbation. Well-posed and ill-posed are two halves of one idea.
The heat IBVP u_t = k u_xx on 0 < x < L with u(0,t)=u(L,t)=0 and u(x,0)=f(x) is well-posed: a solution exists (the Fourier sine series), it is unique (an energy estimate shows two solutions with the same data coincide), and it is stable (each mode only decays, so a small change in f gives a small change in u for all later time).
Well-posed = existence + uniqueness + continuous dependence (stability), all three.
Well-posedness is relative to a choice of data and of solution concept: a problem ill-posed for classical solutions may become well-posed once you broaden the notion of solution (weak solutions) or pose the matching data for the equation's type. It is not a fixed property of the equation alone.