Let $M$ be a compact Riemannian manifold (without boundary), I would like to know under which regularity conditions can we solve the heat equation $$\begin{align} \partial_tu-\Delta u &= f \\ u(\cdot,0) &= u_0(\cdot) \end{align}$$ in $M\times[0,T]$?

I have just read this question which seems to imply that we can solve the equation when $f\in C^{0,0,\alpha}((0,T)\times M,\mathbb{R})$ and $u_0\in C^{2,\alpha}(M,\mathbb{R})$. I would like to know what is the general method of solving when we have this kind of regularity.

The way I learned about solving heat equation in Euclidean domain was by using heat kernel for unbounded domain, and Galerkin approximation on a bounded domain $\Omega$ with boundary. For the Galerkin method, we use the fact that the eigenvectors of the isomorphism $$ -\Delta:H^2(\Omega)\cap H^1_0(\Omega)\to L^2(\Omega) $$ form an orthogonal basis in $L^2(\Omega)$. This seems to rely on the fact that $u\equiv 0$ on $\partial\Omega\times[0,T]$, I don't know if similar result holds on compact manifold without boundary.