Reference request: Long-term behaviour of the heat equation for bounded initial data 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.
 A: First of all, with initial data in $C_b$, classical solution exist, so there is no need for quotation marks. It is easy to see that the convolution of the initial data $f(x)$ with the Gauss–Weierstrass kernel $p_t(x)$ defines the unique bounded classical solution $u(t, x)$.
It is easy to see that $u(t,x) - u(t,y)$ converges to zero as $t \to \infty$: the functions $p_t(\cdot - x) - p_t(\cdot - y)$ converge in $L^1$ to zero as $t \to \infty$. It is only slightly more difficult to see that this convergence is locally uniform with respect to $x$ and $y$. Therefore, your question is equivalent to the following simpler one: for what initial data $f(x)$, the limit of $u(t, 0)$ exists as $t \to \infty$.
I doubt there is a simpler "if and only if" characterization. One can easily provide sufficient conditions, for example it is sufficient to assume that the mean value of $f$ over the ball $B(0, R)$ has a limit as $R \to \infty$ (by the same argument that one uses when proving that Cesàro convergence implies Abel convergence). And one can equally easily construct counterexamples.
A: I doubt that there is a characterization, but one thing I can say is that the complement of the set of such $f$ contains a dense open set.  
Let $U(f)$ be the solution with initial condition $u(0,x) = f(x)$, and
$$G = \{f \in C_b: \lim_{t \to \infty} U(f)(t,\cdot)\ \text{converges uniformly on compact sets}\}$$
Take some $f_0 \in C_b$ such that $U(f_0)(t,0)$ does not converge as $t \to \infty$.  Let $\delta = \limsup_{t \to \infty} U(f_0)(t,0) - \liminf_{t \to \infty} U(f_0)(t,0) > 0$.  Then for any $f \in G$ and any $c \ne 0$, if $\|g- (f + c f_0)\|_\infty < |c|\delta/2$ 
then $$\limsup_{t \infty} U(g)(t,0) - \liminf_{t \to \infty} U(g)(t,0) \ge |c|\delta - 2 \|g-(f+c f_0)\|_\infty  > 0 $$ 
A: 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.
