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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.

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    $\begingroup$ I do not know the literature, but for the interior problem the answer seems to be as follows. There is a unique up to a constant function $f$ in $\Omega$ such that $\partial_\nu f = g$ on $\partial \Omega$, and $\Delta f$ is constant, say $\lambda$, in $\Omega$. Then $v(t, x) = u(t, x) - f(x) + \lambda t$ satisfies the heat equation with the usual Neumann condition $\partial_\nu v = 0$. $\endgroup$ – Mateusz Kwaśnicki Jan 30 at 15:06
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    $\begingroup$ I don't really get your argument. Can you expand it a little bit? Your point is that one can reduce my question to the problem with homogenous Neumann condition, which is well understood? Anyway the Neumann datum $g$ is time dependent, so I don't understand your assertion on the elliptic problem $\endgroup$ – foo90 Jan 30 at 15:19
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    $\begingroup$ The long-time behaviour of the corresponding problem with Dirichlet boundary conditions can be treated by using the theory of vector-valued Laplace transforms, see [Arendt et. al.: "Vector-Valued Laplaced Transforms and Cauchy Problems" (2011), Section 6.3]. Maybe the methods used there can also be employed to study Neumann boundary conditions? $\endgroup$ – Jochen Glueck Jan 30 at 19:07
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    $\begingroup$ @foo90: Ah, I did not notice that $g$ depends on time! Sorry. Yes, the idea was to switch to the homogeneous Neumann condition, but of course this no longer works when $g$ changes in time. $\endgroup$ – Mateusz Kwaśnicki Jan 30 at 19:54

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