Consider the heat equation $$u_t - a\Delta u = f$$ in a bounded domain with some boundary conditions, e.g. Robin BCs, and the initial condition $u|_{t=0}=u_0$.
Are there some results of the following type? If the initial function $u_0$ is a supersolution of the elliptic equation $$-a\Delta u = f$$ then the solution of the heat equation monotonically decreases in time for all $x$.
I've found an article about monotone methods by Sattinger but it concerns strong solutions. Do you know references concerning weak solutions?