Let $M$ be a (compact) Riemannian manifold without boundary. I want to find a reference for the existence result to $$\begin{align} \partial_t u -\Delta u &= f \\ u(\cdot,0)&=u_0 \end{align}$$ assuming the $f$ is nice and the initial $u_0$ is smooth.

This question is motivated by a vague reference from one of the papers I am reading. It is about harmonic flow between compact manifolds (without boundary). A certain part of it cites a theorem from Hamilton's Harmonic maps of manifolds with boundary but I don't see how it applies since we are working on manifold without boundary.

  • $\begingroup$ Manifolds without boundary are a special case of manifolds with boundary, and are usually much easier to deal with. Unless the theorem you want to cite has some special exclusion of the case $\partial M = \emptyset,$ it should apply. $\endgroup$ – Anthony Carapetis Mar 22 '18 at 4:24
  • $\begingroup$ Are you saying that we can apply the result directly? What usually happens to the boundary condition for manifild without boundary then, does it just disappear? $\endgroup$ – BigbearZzz Mar 22 '18 at 11:16
  • $\begingroup$ Assuming the result says something like "For any $u_0 \in C^\infty(M), f \in {\rm Nice}(M), v \in C^\infty(\partial M)$ there is a solution $u$ of the given equation with $u|_{t=0} = u_0$ and $u|_{\partial M} = v$", yes. If the boundary is the empty set, any boundary condition is always vacuously satisfied. $\endgroup$ – Anthony Carapetis Mar 22 '18 at 11:22
  • $\begingroup$ @AnthonyCarapetis Sorry for the late reply. The way I learned to solve heat equation (with $u|_{\partial \Omega}=0$) was by Galerkin approximation, which relies on the orthonormal basis which consists of the eigenvectors of $\Delta$. Does this also work when the boundary is empty? I don't know any result about the spectrum of $\Delta$ on compact manifold. $\endgroup$ – BigbearZzz Mar 29 '18 at 8:41
  • $\begingroup$ The spectral theory on compact manifolds is very similar to that of bounded domains, and should have all the properties you need. If you're interested in elliptic and parabolic equations on manifolds, Taylor's PDE book(s) are quite good. In particular Chapter 8 (in Volume 2) studies the spectrum of $\Delta$ on a compact manifold. $\endgroup$ – Anthony Carapetis Mar 29 '18 at 11:54

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