I have a general parabolic PDE on a compact $C^\infty$ manifold $M$ $\textit{ie}$ a (the?) solution $u$ is a probability density and satisfies the PDE in a weak sense (with $C^\infty_c$ test functions). For example take :
$$ \partial u_t = \Delta u + div(v u) $$ with v smooth, boundary conditions (if there is a boundary) and initial conditions.
I would like to prove the strong maximum principle but for that I need regularity. Do you know some references on this question ? Like Brezis' book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" but for manifolds.
Thank you in advance !