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Let $(M,g)$ be a smooth compact Riemannian manifold without boundary. Then there exists a unique fundamental solution $p(x,y,t)$ on $M \times M \times (0,\infty)$ that is $C^2$ w.r.t. to $x,y$ and $C^1$ w.r.t. $t$ that satisfies $$ (\Delta_x-\partial_t) p=0 \quad p(\cdot,y,t) \to \delta_y \quad \text{as } t \to 0. $$ Furthermore, if $f \in C(M)$ is a continuous function $$ u(x,t)= \int_M p(x,y,t)f(y) d\mu(y) $$ is the unique solution of the heat equation with initial data $u(\cdot,0)=f(\cdot)$. I quote this from Chavel's book "Eigenvalues in Riemannian Geometry".

My question now is that if $f \in L^2(M)$, for example if $f=\chi_D(x)$ for some compact subset with a nice boundary, $$ u(x,t) = \int_D p(x,y,t)d\mu(y) $$ is the unique solution of the heat equation with initial data $u(\cdot,0)=\chi_D(x)$. Certainly it is a solution, but is it unique? So if $v(x,t) \in C^2(M\times M \times (0,\infty))$ solves the heat equation with initial data $v(x,0)=\chi_D(x)$, does it hold $$ v(x,t)= \int_D p(x,y,t)d\mu(y)? $$ I'm not sure if the following works: Since $u(\cdot,0) \equiv v(\cdot,0)$ almost everywhere $$ \int_M (u(x,0) - v(x,0))^2 =0 $$ and $$ \frac{d}{dt} \int_M (u(x,t) - v(x,t))^2 d\mu(x) = \int_M 2(u(x,t) - v(x,t))\frac{d}{dt}(u(x,t) - v(x,t)) d\mu(x)\\ = \int_M 2(u(x,t) - v(x,t))\Delta(u(x,t) - v(x,t)) d\mu(x)\\ = -2\int_M |\nabla (u(x,t) - v(x,t))|^2d\mu(x) \leq 0 $$ (last step by using Green's theorem. So that since the term is positive $$ \int_M (u(x,t) - v(x,t))^2 =0 \quad \text{for all } t\geq 0. $$ But can I apply Green's theorem? I'm not sure how smooth $u(x,t)$ is in $t$ direction.

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  • $\begingroup$ Is you manifold compact? $\endgroup$ Oct 26, 2020 at 20:32
  • $\begingroup$ @ArcticChar Yes, sorry I forgot this condition. $\endgroup$
    – sarusa
    Oct 27, 2020 at 9:47
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    $\begingroup$ To answeer your edited question $u-v$ is smooth whenever $t>0$, so there is no issues in using Green's theorem. $\endgroup$ Oct 27, 2020 at 10:08
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    $\begingroup$ Alright, thank you! Do you know if the approach also works for noncompact manifolds under certain assumptions? $\endgroup$
    – sarusa
    Oct 27, 2020 at 10:10
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    $\begingroup$ You may read the recent paper "On the uniqueness for the heat equation on complete Riemannian manifolds" on arxiv and the referrence therein. $\endgroup$ Oct 27, 2020 at 10:17

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The difference $v - \int_D p ~\mathrm d \mu$ solves the heat equation (since heat equation is linear) and has $0$ initial condition, and so must be equal to the constant $0$ solution by uniqueness.

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  • $\begingroup$ Note that solution to the heat equation is in general not unique if $M$ is noncompact. $\endgroup$ Oct 27, 2020 at 9:53
  • $\begingroup$ @ArcticChar Good to know. Do you know why, intuitively, that might happen? $\endgroup$
    – Keshav
    Oct 27, 2020 at 14:21
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    $\begingroup$ I think the intuition is that uniqueness does not hold: there is a classical example here. If you want uniqueness you need to impose extra assumptions. $\endgroup$ Oct 28, 2020 at 10:52

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