Does solution of heat equation increase in time if the source term is positive?

Question

Let $u_t - \Delta u = f$ hold with $u(0) = u_0 \geq 0$ on a bounded domain where $u_0$ is in $H^1$. We take Dirichlet boundary conditions.

If $f \geq0 $, is it true that $u$ is increasing in time?

Answer 1

No. Consider the equation $$ \begin{cases}u_t-u_{xx}=\sin x,& t>0,\quad0< x<\pi\\ u(x,0)=\sin x, & 0\le x\le\pi\\ u(0,t)=u(\pi,t)=0 & t\ge0. \end{cases} $$ Its solution is $$ u(x,t)=e^{-t}\sin x+\sin x, $$ which is decreasing in time.