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.