Backward ill-posedness does not mean universal nonexistence

Orientation only. Assume Countable Choice and fix $T>0$. The backward heat solution map is unbounded shows that the terminal-to-initial map on the terminal profiles of $C^{4,2}$ interval Dirichlet heat solutions is unbounded with respect to the $L^2(0,\pi )$ norms; this is a statement about continuity and stability, not about existence. The terminal data $\sin (k\cdot )$, for integers $k\ge 1$, do have backward solutions (the modes $u_k$ of The backward heat solution map is unbounded), and so does every finite sine sum, by finite linear combination of these solutions. More generally, whenever a forward interval Dirichlet heat solution $u$ exists on $[0,T]$ with initial profile $f$, its terminal profile $g=u(\cdot ,T)$ admits a backward extension on that interval: the same $u$, with initial profile $f$. This range description asserts no existence for arbitrary initial or terminal data. The failure is therefore that arbitrarily small terminal perturbations can correspond to order-one initial states (The backward heat solution map is unbounded), and on general domains the backward solution operator need not be surjective onto any given data class. No claim of universal nonexistence and no well-posedness claim is made here.

The well-definedness half of the picture is also worth recording: on the $C^{4,2}$ class with zero lateral data the terminal data determine the solution, by Backward uniqueness for the heat equation on a bounded interval, so ill-posedness here is exactly the failure of a uniform norm bound, with the amplification ratio $e^{k^{2}T}$ of the $k$-th sine mode (L2 normalisation of the sine modes on an interval) measuring that failure; no further existence or regularity claim is made.