Given a following PDE system with Neumann condition: $u_{t} - \Delta{u} = f\ $ in $\omega$ ($t > 0$), $\frac{\partial{u}}{\partial{v}} = \phi$ on the boundary of $\omega$ ($t > 0$), and $u = u_{0}$ at $t = 0$.
Now, assume $f$ and $\phi$ is independent of time, and $\int_{\partial\omega} \phi \ ds$ + $\int_{\omega} f\ dx \neq 0$ (so, no steady-state solutions to this PDE). Determine the large-time behavior of solution of the above system (i.e, describe how $u(t,x)$ behaves where $t\rightarrow \infty$).
My attempt: It seems to me that since a steady-state solution doesn't exist, if $u$ is the solution to the PDE system, then $u$ cannot converge to $\overline{u}$ where $\overline{u}$ is the steady-state solution, exponentially fast. But I got stuck from here. Can anyone please help with this problem? Any help would greatly be appreciated.
Physically, there is no steady state because $\int_\Omega u(t,x) dx$ is blowing up (in particular its derivative is a nonzero constant). If you take the derivative of this quantity away from $u_t$, then a steady state exists. So you can describe the solution to your problem as a constant times $t$ plus a process which has a true steady state.
Now prove this. You will need multidimensional integration by parts to do it.