Let $u_t - \Delta u = f$ hold with some appropriate initial condition and Dirichlet boundary condition on some smooth bounded domain. We take $f$ to be negative.
Does anyone know a citation that states the result that the solution is decaying/decreasing in time? I.e. I want an $L^\infty$ decay result of the form
$$|u(t,x)| \leq F(t)$$ where $F$ is decreasing in $t$.
How about multiplying by $u$ and integrating over space?
$$u\partial_{t}u-u\triangle u=uf$$
Integrating shows that:
$$\frac{d}{dt}\int u^{2}+\int |\nabla u|^{2}=\int uf$$
This shows that $u$ should be decreasing. Is this what you wanted?