Let $M$ be a closed Riemannian manifold. If $u \in H^1(0,T;L^2) \cap L^2(0,T;H^1)$ is a weak solution of $$u_t - \Delta u = f$$ $$u(0) =u_0$$ where $f \in L^2(0,T;L^2)$ with $f(t,x) \geq 0$ a.e. and $u_0 \in L^\infty$ with $u_0(x) > 0$ a.e., and we know that the solution $u(t,x) \geq 0$ almost everywhere, do we in fact have that $u(t,x) > 0$ a.e.?
I think this should follow from strong maximum principle but I couldn't find a reference.