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On the paper: Bruce Kleiner, John Lott, Notes on Perelman's papers, Geom. Topol. 12(5): 2587-2855 (2008). DOI: 10.2140/gt.2008.12.2587

The authors wrote the following statement:

enter image description here

(Here $-\Delta$ is the semi-posiitve Laplacian operator.)

It seems that on closed Riemannian manifold $(M,g)$, for any compact interval $[a,b]$, with an initial function $u(b)$, we can always solve the backward heat flow $$\frac{\partial u}{\partial t}=-\Delta u+Ru,$$ where $R$ is a smooth function.

Q Can anyone give a reference about the existence (uniqueness) about the above backward heat flow?

Thanks in advance

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    $\begingroup$ If you reverse in time and take an initial datum $u(b)$, it just becomes a standard heat equation, no? $\endgroup$
    – Leo Moos
    Commented Apr 14, 2021 at 3:06
  • $\begingroup$ @LeoMoos Thanks for the comments. Actually, I want to emphasis that on any compact interval $[a,b]$, we can find a solution with any initial data. Do you know the reference about this? $\endgroup$
    – DLIN
    Commented Apr 14, 2021 at 4:43
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    $\begingroup$ What I meant is that reversing the time for a backward heat equation yields a forward equation. You're working on a closed manifold, so the usual estimates for parabolic PDE would yield long-time existence of your solution. $\endgroup$
    – Leo Moos
    Commented Apr 14, 2021 at 12:54

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