Let us consider the heat equation
\begin{align\*}
 \frac{\partial}{\partial t}u(t,x) & = \Delta u(t,x), \\
 u(0,x) & = f(x)
\end{align\*}
on the whole space $\mathbb{R^d}$. If $f \in L^p := L^p(\mathbb{R}^d)$ for $p \in [1,\infty)$ or if $f \in C\_0(\mathbb{R}^d)$, the long term behaviour of the solution $u(t) := u(t,\cdot)$ is well-known:


* If $f \in L^p$ and $p \in (1,\infty)$, then $u(t)$ converges to $0$ with respect to the $L^p$-norm and also with respect to the $L^\infty$-norm as $ t\to \infty$.
* If $f \in L^1$, then $\|u(t)\|\_{L^1} \to |\int\_{\mathbb{R^d}} f(x) \, dx|$ as $t \to \infty$, and we do not have convergence of $u(t)$ with respect to the $L^1$-norm, in general. However, we still have $\|u(t)\|\_\infty \to 0$.
* If $f \in C\_0(\mathbb{R}^d)$, then also $\|u(t)\|\_\infty \to 0$ as $t \to \infty$.


Now, I am interested in the long-term behaviour of $u(t)$ if $f \in C\_b(\mathbb{R}^d)$, i.e. if $f$ is a bounded continuous function. (*Note:* In this case, it is maybe less clear what we mean by a solution of the heat equation, but we can still define a "solution" $u(t)$ as the convolution of $f$ with the heat kernel).


Uniform convergence of $u(t)$ as $t \to \infty$ is probably too much to expect for many functions $f$, but I would be interested in locally uniform convergence:


**Question.** Is there a characterization of those $f \in C\_b(\mathbb{R}^d)$ for which $u(t)$ converges uniformly on compact sets to a constant function as $t \to \infty$?


I would expect that this has been studied somewhere in the literature, so my question is mainly a reference request. 


**Disclaimer.** I discussed this question with a few experts on the matter, but they did not know the solution nor did they know a reference.



Reference request: Long-term behaviour of the heat equation for bounded initial data
Actually it's quite a subfield in the theory of parabolic equations called stabilization of solutions (for large $t$). I'd recommend to start with the article of V. V. Zhikov, *On the stabilization of solutions of parabolic equations*, Math. USSR-Sb., 33:4 (1977), 519–537, for the exposition as well as for the earlier work references. He proves the related results for the parabolic equations in divergence form. Afaik for the heat equation the condition is indeed the existence of the limit mean value in balls of the initial function, as proved in circa 1960s.


