the backward heat equation

Suppose you photograph a blurred image and ask: what sharp picture, after a known amount of blurring, would produce exactly this? Or you measure a body's temperature now and ask what it was an hour ago. Both are requests to run diffusion backward. The backward heat equation is the heat equation with time reversed, and it is the cautionary tale of PDE theory — a beautifully posed-looking problem that is in fact ill-posed.

Reversing time t -> -t in u_t = k u_xx turns it into u_t = -k u_xx, equivalently asking to find earlier data from later data under ordinary forward diffusion. The trouble is exact and visible in Fourier modes. Forward in time, the mode sin(n x) decays like e^(-k n^2 t); to go backward you must multiply by e^(+k n^2 t), which grows explosively with n. So an imperceptibly small high-frequency wiggle in your late data — even pure measurement noise at the level of 10^(-10) — gets amplified by an astronomical factor and swamps the answer. The solution does not depend continuously on the data: arbitrarily tiny changes in the input produce arbitrarily large changes in the output. By Hadamard's definition that is exactly what ill-posed means.

This is the sharpest illustration that well-posedness depends on the direction and type of a problem, not just on whether a formula exists. The forward heat equation is a model of a well-posed parabolic problem; reverse the clock and the very same equation becomes pathological. It is not merely hard — no amount of cleverness makes it stable. In practice, problems that secretly are backward diffusion (deblurring, thermal back-tracking, source identification) are handled only by regularization: you replace the exact backward problem with a nearby well-posed one by adding assumptions (smoothness bounds, Tikhonov penalties) that tame the explosive high frequencies. Caveat: this does not 'solve' the backward equation; it answers a deliberately modified, stabilized question.