I am interested in the large time asymptotic behavior of parabolic equations. For instance, let $\Omega$ be a regular bounded open subset of $\mathbb{R}^n$ . Let $g \in C(]0,+\infty[ \times \partial\Omega)$. Let $$ \Omega^- \equiv \mathbb{R}^n \setminus \overline \Omega.$$ Consider the following exterior Neumann problem for the heat equation:

\begin{equation} \tag{NP} \begin{cases} \partial_t u -\Delta u = 0 \hspace{1cm} &\mbox{ in } ]0,+\infty[ \times\Omega^-,\\ \partial_\nu u=g &\mbox{ on } ]0,+\infty[ \times\partial\Omega^-,\\ u(0,\cdot) = 0 &\mbox{ in } \overline {\Omega^-}. \end{cases} \end{equation}

I am wondering if there are known conditions on the Neumann datum $g$ which guarantee the physical solution $u$ of problem (NP) (the unique solution under the growth condition in space) to be bounded or to have a limit as $t$ goes to $+\infty$ (by intuition I would say a condition on the integral of $g$). I am interested in the exterior problem, but to start I am also interested in similar results for the interior one. To me it seems a quite classical question, so I am surprised that I couldn't find almost nothing in the literature. For example Friedman in his famous monograph "PDE of Parabolic Type" considers only the large time behavior for the Dirichlet problem and for a specific Robin problem ($\partial_\nu u(t,x)+f(t,x)u(t,x) = g(t,x)$ on $]0,+\infty[\times \partial \Omega$ with $f<-c$ for some c<0) which do not include the Neumann problem.