Let $u : \mathbb R^n \to \mathbb R$, and define $☯^1u : \mathbb R^n \to \mathbb R$ to be the traditional heat transfer of $u$ in a uniform material after $1$ second of time. For example, in this figure we have on the left $u : \mathbb R^2 \to \mathbb R$, and on the right is $☯^1u$.
(Note that ☯ is only defined up to choices of constants but don't let that stop you from solving this problem)
Here is my question. Is it possible to reverse this arrow here? Given a $v : \mathbb R^n \to \mathbb R$, how can we find a $u : \mathbb R^n \to \mathbb R$ and $t \in \mathbb R^{>0}$ such that $☯^tu = v$? By an abuse of notation, what I want to do is receieve a vector $v$ and "compute" $☯^{-s}v$ for $s>0$.
It seems obvious to me, first of all, that no information is actually lost in this process. Also, if a spectrum of functions "heat" to a given input, it's okay. As long as we can choose one particular solution I'm happy. Is there any papers discussing the possibility of this process? I assume the best solution would just be a differential equation which can invert the heat transfer, but if that isn't possible, then I assume some sort of infinitesimal local approach could work where you zoom in enough to where the heat isn't changing by that much in the locally chosen region, solving that, and gluing it all back together in some sort of atlas-like approach.
