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This might be a very simple question, but basically I am looking for a good reference for studying the heat equation on Riemannian manifolds with boundary, specifically when data is put on lateral sides of the boundary. A more precise formulation is given below.

Let $(M,g)$ be a compact smooth Riemannian manifold with a smooth boundary. Let $T>0$, let $V\in C^{\infty}([0,T]\times M)$ and consider the heat equation with boundary data $f$: \begin{equation}\label{pf0} \begin{aligned} \begin{cases} \partial_t u-\Delta_g u+Vu=0\,\quad &\text{on $(0,T)\times M$}, \\ u=f\,\quad &\text{on $\Sigma=(0,T)\times \partial M$,}\\ u(0,x)=0 \,\quad &\text{on $M$,} \end{cases} \end{aligned} \end{equation} I haven't found any references for regularity of solutions to this rather standard PDE with $f$ in Sobolev spaces. Can something precise be said about the regularity of solutions, say if $f \in H^k_0(\Sigma)$ for $k>0$ or perhaps when $f \in H^r_0(0,T;H^s(\partial M))$?

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