I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\partial u}{\partial \nu}=0, & (t,x)\in (0,\infty)\times\partial\Omega\\ 0\leq u(0,x)=u_0(x), & x\in\Omega\end{cases}$ that has a unique bounded solution which is as regular as needed. Moreover $f$ can be taken globally Lipschitz or maybe monotone (if needed) and $u_0\in L^{\infty}(\Omega)$. The domain $\Omega$ is also as smooth as needed.
Suppose also that the bounded function $U$ is the unique steady-state of the above problem, i.e. $\begin{cases} -\Delta U=f(U), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\partial U}{\partial \nu}=0, & (t,x)\in (0,\infty)\times\partial\Omega\\ U\geq 0, & x\in\Omega\end{cases}$
is it true that $\lim\limits_{t\to\infty} u(t,\cdot)=U$ in some sense, say in $L^2(\Omega)$?
Or a more general question: A solution of an evolution problem must be attracted by a steady state knowing that there are steady-states?
