Ok, this is a good point. I should have added a requirement that the true solution is C infinity on the part of the manifold that isn’t in our temporal past. The backward’s heat equation is ill-posed for this reason on...it can’t be propogated arbitrarily far forward (i.e. back).
I disagree. See the backwards heat equation for a simple example:
du/dt = -laplacian(u)
The higher the frequencies you allow in the solution, the faster tiny bits of noise blow up on you.
Ok, this is a good point. I should have added a requirement that the true solution is C infinity on the part of the manifold that isn’t in our temporal past. The backward’s heat equation is ill-posed for this reason on...it can’t be propogated arbitrarily far forward (i.e. back).