# Uniqueness of the solution of the heat equation on a Riemannian manifold

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.

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

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.
• Note that solution to the heat equation is in general not unique if $M$ is noncompact. Oct 27, 2020 at 9:53